Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Doubt in proof of Bertrand Postulate I studied proof of Bertrand Postulate from M Ram Murthy Problems in analytic number theory and completely understood it .
In M Ram Murthy Book , Statement of Bertrand Postulate is (1) - For n sufficiently large , there exists a prime between n and 2n.
But while I was looking at Book... | Statement 1 probably tells you something like "if $n \geq 750$, then there is a prime between $n$ and $2n$". That's the bound that I was taught, where my proof ultimately hit the inequality $\frac{n \log 4}{3} < (2 + \sqrt{2n}) \log(2n)$ that was required to fail.
Now you can obtain statement 2 by checking that there i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3509687",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Is an isolated point in $\mathbb{R}^d$ a limit point? I just read the definition of limit point and isolated point in the book Real Analysis: measure theory, integration and hilbert spaces by e.m. stein and r. shakarchi
A point $x\in\mathbb{R}^d$ is a limit point of $E$ if for every $r>0$, the ball $B_{r}(x)$ contains ... | The way these definitions are written here, yes, it looks like isolated points become limit points.
This is, however, not the conventional definition of limit points. We usually require that $B_r(x)$ contains points of $E$ in addition to $x$ itself. With the conventional definition, isolated points are not limit points... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3509776",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
About an order of a p-group I'm trying to show that if G is a Group, then $|G| = p^2 \Rightarrow G$ is abelian.
The path I'm taking relies on supposing that $|Z(G)| = p$ and forming the quotient group $\overline{G} = G/Z(G)$.
Then we got $\overline{G}$ as a cyclic group, because it has order p. $\overline{G}=<a.Z(G)>,... | It's well known that the center of a $p$-group is nontrivial. This can be seen by looking at the class equation.
The rest goes through without a hitch.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3509896",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 3
} |
Show that $(1+\frac 1n)^{n^2} \mathrm e^{-n}$ is not a zero sequence without logarithms How can I show that
$z_n = \left(1+\dfrac 1n \right)^{n^2} \mathrm e^{-n}$
is not a zero sequence? WolframAlpha says, it converges to $\frac{1}{\sqrt{e}}$ but how can I prove? I'd appreciate a solution without logarithms.
| We will first argue that $z_n \ge e^{-1/2}$, by showing that that for $x \in [0,1], e^{x - x^2/2} \le 1 + x$.
TO do this, let
$$ f(x) := e^{x - x^2/2} - 1 - x.$$ Note that, $$ f'(x) = (1-x) e^{x -x^2/2} - 1, \\ f''(x) = ((1-x)^2 -1)e^{x- x^2/2}$$
Note that the second derivative is non-positive for $x \in [0,1]$. Thu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3510012",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Limit with definite integral
Given
$$f(x)=\int _0^x \dfrac{\sin t}{t} dt$$
calculate
$$\lim _{x \rightarrow 0} \dfrac{2f(x)-f(2x)}{x-f(x)}.$$
I applied L'Hospital's rule so now I have:
$$\lim _{x \rightarrow 0} 2\dfrac{\frac{\sin x}{x}-\frac{\sin 2x}{2x}}{1-\frac{\sin x}{x}}$$
Now I don't know how to proceed.
| Use Taylor's expansion at order $3$ after L'Hospital:
$$ \frac{2\sin x-\sin 2x}{x-\sin x}=\frac{2x-\dfrac{x^3}3-2x+\cfrac{4x^3}3+o(x^3)}{x-x+\cfrac{x^3}6+o(x^3)}=\frac{x^3+o(x^3)}{\cfrac{x^3}6+o(x^3)}=\frac{1+o(1)}{\frac 16+o(1)}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3510144",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Finding linear recurrence for a sequence which depend on another recurrence I stumbled upon a sequence of numbers $b_n$ such that for all $n\in \mathbb{N}$, $$b_n=\sum_{j=0}^M a_{n,j}$$
where $a_{n,j}$ are given by the recurrence,
$$a_{n,j}=\begin{cases}
1 & n=0 &\land &j=0\\
0 & n=0 &\oplus &j=0\\
\sum_{i=0}^M \lamb... | I don't see any immediate way for this to be rewritten into a linear recurrence. However, we can get a closed form. Let us rewrite
$$a_n=\begin{bmatrix}a_{n,0}\\a_{n,1}\\\vdots\\a_{n,M}\end{bmatrix}$$
$$\lambda=\begin{bmatrix}\lambda_{0,0}&\lambda_{1,0}&\cdots&\lambda_{M,0}\\\lambda_{0,1}&\lambda_{1,1}&\cdots&\lambda_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3510263",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Length of cable parabola with height $\propto$ (horizontal distance from midpoint)² Question:
The cable to a certain suspension bridge has the shape of a parabola. The height at a certain point is proportional to the square of the horizontal distance from the midpoint. The dimensions are given in the image.
Determine t... | Your formula, since $h(0)=h_0$, is $h(x)=kx^2+h_0$. And the $h_0$ plays no role in the length formula, because the derivative will kill it. Then your integral gives the value you want.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3510431",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Eigenvectors and Eigenvalues of Shift Matrix $$S:\mathbb{C}^n\rightarrow\mathbb{C}^n, $$
$$S(x_1,x_2,...,x_n)^T = (x_n,x_1,...,x_{n-1})^T.$$
How can the eigenvalues and eigenvectors of S be calculated?
I already have the standard matrix of S which is:
\begin{bmatrix}
0 & 0 & 0 & \dots & 0 & 1\\
1 & 0 & 0 & \d... | To find its eigenvalues, note that $\lambda I - S$ is given by the following matrix:
\begin{eqnarray}
\begin{pmatrix}
\lambda & 0 & 0 & \cdots & 0 & -1 \\
-1 & \lambda & 0 &\cdots & 0 & 0 \\
0 & -1 & \lambda & \cdots & 0 & 0 \\
0 & 0 & -1 &\cdots & 0 & 0\\
\vdots & \vdots & \vdots & \ddots& \vdots & \vdots \\
0 & 0 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3510533",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Homotopy of functions of $S^1$ and $\mathbb{R}P^2$ So I was doing an exercise that asked me that if there exist two functions $g \colon S^1 \rightarrow \mathbb{R}P^2$ and $f\colon\mathbb{R}P^2 \rightarrow S^1 $ then $f \circ g$ is homotopic to the identity in $S^1$. I think this is false and my argument is as follows. ... | I think your argument is correct, but it can be expressed more succinctly using the functoriality of $\pi_1$.
You can show that any continuous composition $S^1 \stackrel{g}{\to} \mathbb{R}P^2 \stackrel{f}{\to} S^1$ is null-homotopic by considering the composition $\pi_1(S^1) \stackrel{\pi_1(g)}{\to} \pi_1(\mathbb{R}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3510660",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
} |
Determinant of a particular type of matrix I was doing a problem and found that if I could get the determinant of this matrix, it would make solution easier. Eventually, I gave up and solved it another way. I am still curious as to how I would go about calculating the determinant of this $n \times n$ matrix:
$$A=\begin... | \begin{eqnarray*}
A=\begin{bmatrix}
a & 0 & \ldots & 0 & -a\\
0 & a & \ldots & 0 & -a\\
\vdots & \vdots & \ddots & \vdots & \vdots\\
0 &0 & \ldots & a & -a \\
-a & -a & \ldots & -a & b
\end{bmatrix}
\end{eqnarray*}
Add each column to the last column
\begin{eqnarray*}
A=\begin{bmatrix}
a & 0 & \ldots & 0 & 0\\
0 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3510823",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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What assumption on $x$ is needed for $x x^\top$ to be positive definite? Consider $M = x x^\top$, $x \in \mathbb{R}^n$
For $n = 1$: $M = x^2$, then $M$ is PD if $x \neq 0$.
For $n = 2$, $M = \begin{bmatrix} x_1^2 & x_1x_2 \\ x_2x_1 & x_2^2 \end{bmatrix}$, which is PD when $x_1 \neq x_2$ and $x_1 \neq 0 \wedge x_2 \neq... | Observe that for any vector $v\in\mathbb R^n$ we have that
$$
v^TMv=(v^Tx)(x^Tv)=\langle v,x\rangle^2.
$$
For $M$ to be positive definite, we need this to be strictly greater than zero for all non-zero $v$. But this is impossible if $n> 1$, since the orthogonal complement of the subspace spanned by $x$ is non-empty.
In... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3510966",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Do arbitrary union and arbitrary intersection operations commute? Let $X$ be a set, and $\mathcal{S}$ a set of subsets of $X$. Let $\sigma(\mathcal{S})$ be the set of all arbitrary unions of the elements of $\mathcal{S}$, and $\tau(\sigma(\mathcal{S}))$ the set of all arbitrary intersections (empty intersection being $... | Consider that for subsets $A_{i,j} \subseteq X$ and index set $J$ (and for each $j \in J$ and index set $I_j$), we can rewrite a member of $\tau(\sigma(\mathcal{S}))$ as:
$$\bigcap_{j \in J} \bigcup_{i \in I_j} A_{i,j} = \bigcup_{f \in F} \bigcap_{j \in J} A_{j, f(j)}$$
where $F$ is the set of all functions $f$ from $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3511075",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
How do I rotate a square around a circle? I have
*
*a circle of radius r
*a square of length l
The centre of the square is currently rotating around the circle in a path described by a circle of radius $(r + \frac{l}{2})$
However, the square overlaps the circle at e.g. 45°. I do not want the square to overlap wi... | Consider what happens when the square is directly to the right of the circle with the circle tangent to the midpoint of a side of the square.
In order for the square to move up while remaining in contact without crossing into the square, it has to go straight up.
This continues until the lower left vertex is on the c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3511223",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "40",
"answer_count": 4,
"answer_id": 1
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Use dominated convergence theorem to show: $\lim_{\ n \to \infty} \| f \ 1_{[n,n+1]} \|_1 = 0$ Consider the Lebesgue measure $\lambda$ on $\mathbb{R}$.
Given that $f: \mathbb{R} \to \mathbb{R} $ is integrable,
Ik would like to show the following using the dominant convergence theorem:
$$\lim_{\ n \to \infty} \| f \ 1_{... | $$
\int\limits_{\mathbb R} f\,d\lambda = \sum_{n\,\in\,\mathbb Z} \,\,\, \int\limits_{\mathbb R} f\mathbf 1_{[n,n+1]} \, d\lambda
$$
The series converges absolutely because $\|f\|_1<\infty.$ The terms of a convergent series approach $0.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3511365",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Maximum number of students such that they all have at least 3 out of 6 different answers. A teacher made a test for his mathematics class with 6 true or false questions. When he received the tests, he noticed that any pair of students had at least three diferent answers. Since all the students answered every question, ... | To see that $9$ is impossible:
Note that if you only had $4$ questions you could have no more than $2$ students whose pairwise answers differed by at least three questions. Indeed, pick one of the students. Relabeling if necessary we can assume that this student chose $TTTT$. Any other student would need to have cho... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3511773",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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Show: For every $\alpha \in \mathbb{Q}$, the polynomial $X^2+\alpha$ has endless different roots in $\mathbb{Q}^{2\times2}$.
Show that for every $\alpha \in \mathbb{Q}$, the polynomial $X^2+\alpha$ has endless different roots in $\mathbb{Q}^{2\times2}$.
What I did:
Let $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix... | HINT: What happens if $a=-d$ and $bc=-a^2-\alpha$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3511852",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Calculating Volume of the Submerged Portion of a Rotated Rectangular Prism I'm trying to simulate buoyancy forces on rectangular prism of arbitrary orientation. Suppose that the prism has size $L\times W\times H$ and an Euler angle of $(\phi, \theta, \psi)$. Furthermore, as a rudimentary approximation, the water can be... | Answer: $V = LW(d_y + H/2)$, where $d_y$ is the $y$-coordinate at which the surface of the water hits the $y$-axis.
If $\theta$ is the angle that the $y$-axis of the prism makes with the vector pointing straight up, then $d_y = h/\cos \theta$.
Note: This formula fails when the prism lies such that the $y$-axis is paral... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3512057",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Show that: $\lim_{n\rightarrow\infty}e^{\frac{\log x}{\log\log xn-\log\log n}-\log\left(n\right)}=\sqrt{x}$. I want to show that:
$$\lim_{n\rightarrow\infty}e^{\frac{\log x}{\log\log xn-\log\log n}-\log\left(n\right)}=\sqrt{x}$$
I looked it up on Wolfram Alpha, and it says:
$$\lim_{n\rightarrow\infty}e^{\frac{\log x}{\... | Yes, you are correct. If $x>0$ then the limit is $\sqrt{x}$.
Note that as $n\to +\infty$,
\begin{align}\log(\log(nx))&=\log\left(\log(n)\left(1+\frac{\log(x)}{\log(n)}\right)\right)\\
&=\log(\log(n))+\frac{\log(x)}{\log(n)}-\frac{1}{2}\frac{\log^2(x)}{\log^2(n)}+o(1/\log^2(n)).
\end{align}
Hence
\begin{align}\frac{\lo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3512202",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Why is it that given $E=\{ 1, \{2,3\}, \{2,4\} \}$ we have $1\in E$ but $2\not\in E$? I am reading "Book of proof" of Richard Hammard and it says that given the set $E=\{ 1, \{2,3\}, \{2,4\} \}$, we have that $2\not\in E$ . I do not understand why is it so. I understand that the set $E$ is a collection of the elements ... | I think this is a deliberately provocative example, chosen by the author to illustrate the thought, that the relation, $\in$ is 'blind' to whatever structure the elements may have. The elements of a set are just anonymous 'lumps' - labels, if you will - and we don't care about what these labels mean, if indeed they mea... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3512341",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Adjoint(s) to the forgetful functor $U:A/\mathbf{C}\to \mathbf{C}$. I am preparing for my exam in Category Theory, and came across the following exercise in an old exam. Let $\mathbf{C}$ a category with finite coproducts. For a fixed object $A$, consider the coslice category consisting of objects $f:A\to C$. Morphisms ... | For the fact that it admits a left adjoint your (not unfounded at all) discussion gives you the awnser (the only problem is you were trying the wrong side) :
$$
\text{Hom}_{\mathbf{C}}(D, U(f:A \to C)) \cong \text{Hom}_{A/\mathbf{C}}(i_A : A \to A \sqcup D, f:A\to C).
$$
Indeed having a map $g : D \to C$ will give, by ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3512488",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Proving that function is continuous and differentiable at $x=0$ I have the following function:
$$
f(x)=\begin{cases}
|x|^{m}\sin\left(\frac{1}{|x|^{n}}\right) & x\neq0
\\
0 & x=0
\end{cases}
$$
I'm trying to find for which $n,m$:
*
*The function $f$ is continuous at $0$.
*The function $f$ is differentiable at $0$... | *
*As you said, $ -1\le\sin(x)\le 1$, hence you can write:
$$|x|^m \sin\Big(\frac{1}{|x|^n}\Big)\le |x|^m $$
So for each $m>0$ the function is continuous. Fo $m=0$ the sin has no limit except in the trivial case $n=m=0$ for which our function is just the constant $\sin(1)$ everywhere but in $x=0$. Note that even in t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3512648",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Calculate $\int_{0}^{\pi/2} (\sin(2x))^5 dx$ I want to calculate the following integral:
$$\int_{0}^{\pi/2} (\sin(2x))^5 dx$$I tried integration by parts but it didn't work for me. Suggestions?
| $$\dfrac{d(\sin^nax\cos ax)}{dx}=an\sin^{n-1}ax(1-\sin^2ax)-a\sin^{n+1}x$$
Integrate both sides with respect to $x,$
$$\sin^nax\cos ax=anI_{n+1}-a(n+1)I_{n+1}$$
where $$I_n=\int\sin^max\ dx$$
Set $n+1=5,3,1$ one by one
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3512754",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 2
} |
Metric on the embedded hyperboloid in five-dimensional Minkowski space I have an explicit parametrisation of a one-sheeted hyperboloid in the five-dimensional Minkowski space, namely:
$$Z_{0}=-l \cdot \cot(\tau)$$
$$Z_{a}=\frac{l}{\sin(\tau)}\omega_{a}, \quad a=1,\ldots,4$$
Where the $\omega_{a}$ represent the coordina... | Let us set $l=1$. One can reinsert the right power of $l$ at the end, as it is nothing but a scaling constant.
Let me first correct that the problem actually concerns the hyperboloid with one sheet $M$ of all unit spacelike vectors in Minkowski $5$-dimensional spacetime, namely the vectors $Z^a$ satisfying $Z^a Z_a = 1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3512876",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
What is the error calculating the sum of this series? I need to determine if $$\sum_{n=1}^\infty \frac{2^n-1}{4^n}$$ converges, in which case I must also find its sum, or diverges. This is my approach:
$i$) $$\frac{2^n-1}{4^n}=\frac{2^n}{4^n}-\frac{1}{4^n}=\frac{2^n}{2^{2n}}-\frac{1}{4^n}=\frac{1}{2^n}-\frac{1}{4^n}$$
... | $\sum_\limits{n=1}^{\infty} k^n = \frac {k}{1-k}\\
\sum_\limits{n=1}^{\infty} (\frac {1}{2})^n = \frac {\frac 12}{1-\frac 12} = 1\\
\sum_\limits{n=1}^{\infty} (\frac {1}{4})^n = \frac {\frac 14}{1-\frac 14} = \frac {1}{3}\\
\sum_\limits{n=1}^{\infty} (\frac {1}{2})^n - \sum_\limits{n=1}^{\infty} (\frac {1}{4})^n = \fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3512982",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Unexpected appearances of $\pi^2 /~6$.
"The number $\frac 16 \pi^2$ turns up surprisingly often and frequently in unexpected places." - Julian Havil, Gamma: Exploring Euler's Constant.
It is well-known, especially in 'pop math,' that
$$\zeta(2)=\frac1{1^2}+\frac1{2^2}+\frac1{3^2}+\cdots = \frac{\pi^2}{6}.$$
Euler's ... | Consider the following picture:, centered at the origin of $\mathbf{R}^{2}.$ It is a concentric arrangement of circles $\color{red}{\text{(- this should be discs ?)}}$; each circle has radius $1/n.$ We can think of it as an infinite bulls-eye. The sum of the areas shaded in red is equal to $\frac{1}{2}\pi\zeta(2).$ In ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3513089",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "45",
"answer_count": 28,
"answer_id": 2
} |
Using the complex definition of $\sin$ to solve an integral Is it possible to use the complex definition of trigonometric identities to simplify integrals? For instance, the integral: $$\int e^{-at}\cos(bt) \, dt$$
Would it be appropriate to use the definition of complex sine and then convert it back after the integrat... | This does not add anything to @Michael Hardy's answer but shows the ooold way I learnt !
Consider $$I= \int e^{-at} \cos(bt) \,dt \qquad \text{and} \qquad J=\int e^{-at} \sin(bt) \,dt$$ So
$$I+iJ=\int e^{-(a+ib)t} \,dt\qquad \text{and} \qquad I-iJ=\int e^{-(a-ib)t} \,dt$$ that is to say
$$I+iJ=-\frac {e^{-(a+ib)t}}{a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3513117",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Radius of convergence of a complex series I need to find the radius of convergence of the following series $$\displaystyle \sum_{n=0}^{\infty} \Big(\frac{z}{1+z}\Big)^n.$$
Here is my solution:
I know that $\displaystyle \sum_{n=0}^{\infty} w^n$ converges absolutely for $|w| < 1$, converges uniformly for $|w| \leq \del... | Your main error is assuming that the area of convergence is a disk (with the border being 'unknown'), by insisting on a convergence radius!
That's only true for a power series, which has the form $\sum_{n=0}^{\infty}a_nz^n$, which your series is not. Other kinds of serieses have other shapes of their area of (absolute)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3513242",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Infinite product with zero value For an infinite product $\prod a_k$ to converge we need
*
*at most finitely many zero factor, let be $m$ the maximum index of them
*$c=\lim_{n\to \infty}\prod_{k=m+1}^n a_k$ must exists, and
*$c\ne 0$.
My question is "Why the additional condition 3?"
Consider
$$\tag{1}
\prod_... | Yes, this is the definition in all books covering infinite products. We say $\prod \frac{n}{n+1}$ "diverges to $0$", and is not included when we say an infinite product "converges". The reason for this definition is that it is useful, for example in complex analysis.
One example: (there are many others)
$$
\sin z= z... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3513417",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Calculate a limit with an integral within proving that it's possible to use the L'Hôpital rule Let $f: [1, +\infty) \rightarrow R\;$ be a continuous function, bounded, and such that $f(x) \ge1 \;\;\;\forall\;x\ge1$. Calculate reasonably the following limit, proving that it is possible to use L'Hôpital Rule:
$$\lim_{x\t... | Note that the denominator $x$ here tends to $\infty $ and thus L'Hospital's Rule can be applied. One should remember L'Hospital's Rule can be applied on two forms: "$0/0$" and "$\text{anything} /(\pm\infty) $".
Applying the rule here we see that limit in question is equal to the limit of $$\frac{f(x^2)}{x^2}\cdot 2x=2\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3513587",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
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How to determine all $2 \times 2$ normal matrices?
Determine all $2 \times 2$ normal matrices.
In particular, how would I show that there are normal matrices which are neither unitary, Hermitian, skew-Hermitian, symmetric, nor skew-symmetric. The only thing I do know is $AA^* = A^*A$, but I'm not sure how to proceed... | Note that $A$ is normal if and only if $AA^* - A^*A = 0$. If we take
$$
A = \pmatrix{a&b\\c&d},
$$
then we find
$$
AA^* - A^*A =
\pmatrix{|b|^2 - |c|^2 & -b \bar a + a \bar c + b \bar d - d \bar c\\
\overline{-b \bar a + a \bar c + b \bar d - d \bar c} & |c|^2 - |b|^2}.
$$
$A$ will be normal if and only if all the ab... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3513736",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Prove the positive sequence $a_{n+1} = \sqrt{1+\frac{a^2_n}{4}} $ is strictly increasing for $0 \leq a_0<\frac{2}{\sqrt{3}}$ My attempt:
$a_{n+1} - a_n= \sqrt{1+\frac{a^2_n}{4}} -a_n > \frac{a_n}{2}-a_n = -\frac{1}{2}a_n$, which doesn't tell me any thing. How do I prove that this sequence is strickly increasing? Than... | HINT:
It remains to show that
$$
\forall n\, a_n<\frac2{\sqrt{3}}.
$$
For $n=1$ it is in comments.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3513861",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Unexpected use of polynomials in combinatorics Can someone please post some (relatively easy, say high school level) combinatorial problems which can be solved with polynomials but NOT generating functions.
Related to this post in ME.SE.
| Here is an example I was talking about:
We have $2n$ different numbers $a_1,...a_n, b_1,...b_n$. A table $n\times n$ is divided on $n^2$ unit cells and in cell $(i,j)$ we write a number $a_i+b_j$. Suppose that all products of numbers written in cells in each column are the same. Prove that then all products of numbers ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3514004",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 9,
"answer_id": 1
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Proving if the set is a group under the following operation So I’ve started learning about group theory this semester (actually just this week) and I’m a total newbie in the field. I’ve got the following set and I need to prove if it’s a group with respect to the operation stated below:
$$ G := \mathbb R \setminus \{... | I think an easy way to prove the closure property is to see that, $a\circ b \ne 1, \forall a, b \in \mathbb R \setminus \{1\}$.
Using the identity $a+b-ab-1=(1-b)(a-1)$, we can see that $a+b-ab \ne 1 \Leftrightarrow (1-b)(a-1) \ne 0, \forall a, b \in \mathbb R \setminus \{1\}$
The inverse of $a$ goes for resolving $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3514088",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Sum of combinations formula Is there an explicit formula for the sum $0\dbinom{n}{0}+1\dbinom{n}{1}+\dots+n\dbinom{n}{n} = \sum_{k=0}^nk\dbinom{n}{k}$?
| Actually, I think I got it because I realized I forgot to show what work I have already done in the question but as I was trying to refine my work I realized what I need to do:
$$0\dbinom{n}{0}+1\dbinom{n}{1}+\dots+n\dbinom{n}{n}=$$
$$1\dbinom{n}{1}+2\dbinom{n}{2}+\dots+n\dbinom{n}{n}=$$
$$1\cdot\dfrac{n!}{1!\cdot (n-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3514166",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
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Is sum of two uniform random variables is uniformly distributed? For example, $X_1, X_2 \sim U[0,t]$. Does it imply that $2X_1+X_2 \sim U[0,3t]$?
| If $X_2$ is always equal to $X_1,$ and $X_1\sim\operatorname{Uniform}[0,t],$ then $2X_1+X_2\sim\operatorname{Uniform}[0,3t].$ At the opposite extreme, if $X_1,X_2\sim\operatorname{Uniform}[0,t]$ and $X_1,X_2$ are independent, then $2X_1+X_2$ is not uniformly distributed. You haven't told use the joint distribution of $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3514321",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Using Beta Gamma function, show that :$\int_0^{\frac {\pi}{6}} \cos^2 (6\theta).\sin^4 (3\theta) d\theta$ Using Beta Gamma function, show that :$\int_0^{\frac {\pi}{6}} \cos^2 (6\theta)\cdot\sin^4 (3\theta) d\theta$
My Attempt
$$\int_0^{\frac {\pi}{6}} \cos^2 (6\theta) \cdot \sin^4 (3\theta) d\theta$$
Put $3\theta=t$
... | Hint:
Use $\cos2t=1-2\sin^2t$
$$\cos^2(2t)=(1-2\sin^2t)^2=?$$
$$\beta\left(m,n\right)=2\int_0^{\pi/2}\sin^{2m-1}t\cos^{2n-1}t\ dt$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3514501",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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$(A=A\cup B )\Leftrightarrow B\subset A$ - is it true? I'm trying to find out whether the following statement is true:
$(A=A\cup B )\Leftrightarrow B\subset A$
In my opinion it is, because $A=A\cup B$ if B is either an empty set, or $A=B$ or $B \subset A$, all of which are are equivalent with $B \subset A$.
Could you p... | $$B\not\subseteq A\iff \exists x\in B\setminus A\iff x\in A\cup B, x\notin A \iff A\neq A\cup B$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3514781",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Intersection of union of measurable set Let $f_n$ be a sequence of measurable functions on $\mathbb R$.
Show that the set $A:= \{x \in \mathbb R \mid f_n > 0\text{ for infinitely many } n \}$ is measurable.
If I write the set $A$ like intersection of the union of a measurable set, then I am done.
But I can not.
Ple... | Let $$E_n:= \{f_n > 0\}=\{x\in\mathbb R\mid f_n(x)>0\}$$
$$A = \bigcap_{n=1}^\infty \bigcup_{k=n}^\infty E_n$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3514917",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
} |
Given the premises p→q and ¬p→¬q, prove that p is logically equivalent to q Given the premises p→q and ¬p→¬q, prove that p is logically equivalent to q.
I understand why this works, but I do not know how to construct a complete formal proof. So far, I have this:
premises:
p→q ¬p→¬q
(p→q) ∧ (¬p→¬q) ⇔ T
apply contraposi... | Hint:
To prove this use natural deduction, in general we will use $\leftrightarrow\text{ Intro }$, so we want to first assume $Q$, see if this indeed implies $P$, another direction is trivial by $\to\text{ Elim }$ from $P\to Q$.
Answer:
$$\def\fitch#1#2{\quad\begin{array}{|l}#1\\\hline#2\end{array}}\fitch{1.~P\to Q\\2.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3515063",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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if $w$ is a complex number, how to show that $w^\left(1/2\right)$ has 2 roots? except the case for $w=0$ I am looking for a convincing argument to show that if $w$ is a complex number, $w^{\frac 12}$ has 2 different roots.
| Consider the polar representation $w=r e^{i(\phi+2k\pi)}$.
Its square root yields 2 distinct roots that repeat ad infinitum.
That is:
$$w^{1/2}=\sqrt r e^{i(\phi/2+k\pi)}$$
If $r=0$ we have only 1 root, which is 0. Otherwise we get the 2 distinct roots $\sqrt r e^{i\phi/2}$ and $\sqrt r e^{i(\phi/2+\pi)}$. All other va... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3515241",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Does $f(x) = x - \tanh(x)$ have an inverse function that can be expressed in terms of elementary functions? I find this question relevant in my current study of the tractrix, namely because this expression appears in one parameterization of the curve. I’ve noticed that the plot of the Cartesian equation of the tractrix... | Inverse functions are sometimes transcendental in nature, cannot be expressed in terms of elementary functions.
Way to derive 3D coordinates of asymptotic lines on a pseudosphere for tractrix , in polar/cylindrical coordinates $(r,\theta,z$ ):
$$ \sin \psi = \sin \phi = r/a = \frac{r d\theta}{ds}$$
$$ dr/ds= \sin \phi ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3515440",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Why graphs are so important The notion of graph seems to be of a huge importance in Computer Science. A very rich Graph Theory was developed, and theoretical problems are being solved all the time.
What I don't understand, is what makes graph objects so important? Which intrinsic properties of graphs make them so usefu... | Graphs are a common method to visually illustrate relationships in the data. The purpose of a graph is to present data that are too numerous or complicated to be described adequately in the text and in less space.
Wikipedia says,
Graphs can be used to model many types of relations and processes in physical, biological,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3515632",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Vector Space is a Free Module
Is this argument correct? They assume $B$ doesn’t span $V$. Then ${B}^{‘} = B \cup (v) $. And they say $\alpha v + \sum_{i=1}^r \alpha_ib_i$ = 0 implies that $\alpha\neq0$ otherwise every scalar becomes zero. My question here is that there is typo here isn’t it? $B^‘$ must be linearly dep... | $v$ is by assumption (that $B$ is not a basis) not a linear combination of any finite subset of $B$. In particular, $v \notin \text{span}(B)$. So it's right, that $B' = B \cup \{v\}$ is linear independent (because if it was linear dependent, then there would be an $(\alpha,\alpha_1,...,\alpha_r) \neq 0$ and a sum $\alp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3515995",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Hall and Knight question If n is any positive integer show that the integral part of $$(3+\sqrt7)^n$$is a odd number
I have no idea how to begin this problem but it is given in the chapter of binomial theorem so I hope that it is found using that only
| 'I'to denote the integral and 'f'to denote the fractional part of $(3+√7)^n$
Now $(3-√7)^n$ is less than 1 and a proper fraction let's
denote it by f'
$(3+√7)^n=3^n+ C_13^{n-1}√7......$
$3-√7)^n=3^n-C_13^{n-1}√7........$
As you can see when we add them the irrational terms cancel out.
$(3+√7)^n+(3-√7)^n$=I+f+f'= even... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3516145",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Can the quadratic formula be explained intuitively? Most people know what the quadratic formula is so I won’t post it here (Besides, I don’t know how to properly post formulas in general).
I was wondering if there is an intuitive explanation as to why the quadratic formula is structured the way it is.
| The equation $$ax^2+bx+c=0$$ can be normalized to a simpler form by using a linear change of variable such as
$$x=pt+q.$$
Plugging in the equation, we get
$$ap^2t^2+(2apq+bp)t+aq^2+bq+c=0.$$
Now (WLOG $a>0$) we are free to set
$$\begin{cases}ap^2=1,\\2apq+bp=0\end{cases}$$
and the equation simplifies to
$$\color{green}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3516253",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is the following true: $[\nabla \times(\vec{A} \times \vec{B})] \cdot \vec{C}=(\vec{C} \times \vec{\nabla}) \cdot(\vec{A} \times \vec{B})$? Is the following equation correct? $$[\nabla \times(\vec{A} \times \vec{B})] \cdot \vec{C}=(\vec{C} \times \vec{\nabla}) \cdot(\vec{A} \times \vec{B})$$
If so, how can this be show... | Having $\vec A\times \vec B$ just muddies the waters. Just try to see instead why
$$(\nabla\times\vec A)\cdot\vec C = (\vec C\times\nabla)\cdot\vec A$$
for any vector fields $\vec A$ and $\vec C$.
Let's write out the right-hand side in terms of components. Writing $\vec A = (A_1,A_2,A_3)$ and $\vec C = (C_1,C_2,C_3)$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3516430",
"timestamp": "2023-03-29T00:00:00",
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High school math books recommendations I'm in last middle school year and want to learn High School maths in depth — I want to grasp the behind the scenes and learn proofs. What are good books to self study HS maths this way? In what order should I study?
| I add to the answer of Chris a very nice book in English language. I have discovered this morning that some examples are taken from my textbook in italian language.
The name of the book it is the necklace of JAMES STEWART
Here there is a preview of Algebra: https://www.stewartcalculus.com/data/CALCULUS_8E_ET/upfiles/6... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3516516",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to find the matrix A with the given linear transformation? I'm working on an assignment and I came across this problem and I am not really sure how to approach it. Any advice would be really helpful.
Find an example of a linear transformation $T:\mathbb{R}^2\to\mathbb{R}^3$ given by $T(x)=Ax$ such that
$T([1,1]) = ... | Why not let
$$A=\begin{bmatrix} 3 & 0 \\ 3 & 0 \\ 5 & 0 \end{bmatrix}$$
Then you'd have that
$$T(x)=Ax=\begin{bmatrix} 3 & 0 \\ 3 & 0 \\ 5 & 0 \end{bmatrix}\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}=\begin{bmatrix} 3x_1 \\ 3x_1 \\ 5x_1 \end{bmatrix}.$$
And it would follow easily that $T\left(\begin{bmatrix}1\\1\end{bma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3516607",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How do I find $\lim_{x \to 8} \frac{(\sqrt[3]{x} -2)}{x-8}$ by using the conjugate rule? I need to find: $\lim_{x \to 8} \frac{(\sqrt[3]{x} -2)}{x-8}$
I cannot solve this by substitution because that would cause the denominator to equal 0.
Normally, I would simply use the conjugate trick, however I am uncertain how I ... | Take the steps below$$\lim_{x \to 8} \frac{\sqrt[3]{x} -2}{x-8}
=\lim_{x \to 8} \frac{(\sqrt[3]{x} -2)((\sqrt[3]{x})^2 +2\sqrt[3]{x} + 4)}{(x-8)((\sqrt[3]{x})^2 +2\sqrt[3]{x} + 4)}$$
$$=\lim_{x \to 8}\frac{x-8}{(x-8)((\sqrt[3]{x})^2 +2\sqrt[3]{x} + 4)}
=\lim_{x \to 8}\frac{1}{(\sqrt[3]{x})^2 +2\sqrt[3]{x} + 4}
=\frac1{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3516838",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is there a general formula for harmonic number at present? New to stackexchange.
Known harmonic numbers are defined as
$$ H_n= \sum _ {i = 1}^n\frac {1} {i}$$
Is the series above similar to the general term formula of $ \sum _ {i = 1}^n i= n (n+1)/2$?
|
The method is the same as for $\sum_{j \le n} j^r$ but the result isn't a polynomial, we only have an asymptotic expansion.
Let $$f(z) = \frac1z - \log(z+1)+\log(z)=\frac1z - \log(1+\frac1{z})=F(\frac1z)$$
$F(s)= s-\log(1+s)$ is analytic for $|s|<1$ and $F(0)=F'(0)=0$. Thus, by induction there are some coefficients $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3516956",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Knowing that $\prod_{i = 1}^na_i = 1$, prove that $\prod_{i = 1}^n(a_i + 1)^{i + 1} > (n + 1)^{n + 1}$.
Given natural $n$ $(n \ge 3)$ and positives $a_1, a_2, \cdots, a_{n - 1}, a_n$ such that $\displaystyle \prod_{i = 1}^na_i = 1$, prove that $$\large \prod_{i = 1}^n(a_i + 1)^{i + 1} > (n + 1)^{n + 1}$$
We have that... | For the numbers $2^{\frac{n}{2}}$ is greater than n+1 obviously satisfies, for n>5 can be proven by induction (hint: x$\sqrt{2}$ - x - 1>0 for x> $\sqrt{2}$ + 1)and for other n<6 check manually
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3517123",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the limit $\lim_{x\to 0} x\left(\left[\frac{1}{x}\right] +\left[\frac{2}{x}\right] +\cdots \left[\frac{10}{x}\right] \right)$ Can someone help me finding the following limit
$$ \lim_{x\to 0} x\left(\left\lfloor\frac{1}{x}\right\rfloor +\left\lfloor\frac{2}{x}\right\rfloor +\cdots \left\lfloor\frac{10}{x}\right\rf... | $$1>\frac{i}{x}-\left[\frac{i}{x}\right]\ge0$$
For $x>0$, multiply by $x$
$$x>i-x\left[\frac{i}{x}\right]\ge0$$
Sum for $1\le i\le10$
$$10x>55-x\sum \left[\frac{i}{x}\right]\ge0$$
Let $x$ tend to $0$, then $x\sum \left[\frac{i}{x}\right]$ tends to $55$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3517293",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 3
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Inverse cdf of the $\chi$-squared distribution I need to evaluate the functional inverse of the $\text{cdf}$ of the $\chi$-squared distribution
$$\text{cdf}_{\chi_\nu}(t)=\mathbb P(X^2_\nu>t)=\frac1{2^\nu\Gamma(\frac\nu2)}\int_0^te^{-x^2}x^{\nu/2-1}dx.$$
The value of $t$ is fixed (say $0.9$), but the number of degrees ... | I have found relevant information in this paper: "Exploring How to Simply Approximate the P-value of a Chi-Squared Statistic, Eric J. Beh, Austrian Journal of Statistics
June 2018, Volume 47, 63-75."
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3517407",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Integration inequality from ISI Entrance Examination Let $f(x)$ be a continuous function, whose first and second derivatives are continuous on $[0,2\pi]$ and $f''(x)\ge 0$ for all $x\in [0,2\pi]$. Show that $$\int_0^{2\pi}f(x)\cos x dx\ge 0$$.
| Let us complete your attempt.
According to the mean value theorems for definite integrals there is a $c\in (0,2\pi)$ such that
\begin{equation*}
\int_{0}^{2\pi}f''(x)\cos x\, \mathrm{d}x = \cos(c)\cdot\int_{0}^{2\pi}f''(x)\, \mathrm{d}x = \cos(c)(f'(2\pi)-f'(0)).
\end{equation*}
Consequently
\begin{equation*}
I= (f'(... | {
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"url": "https://math.stackexchange.com/questions/3517544",
"timestamp": "2023-03-29T00:00:00",
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Continuous in subspace topology Suppose $X = \lbrace 1,2,3,4,5 \rbrace$ and $\tau = \lbrace X, \emptyset , \lbrace 1 \rbrace , \lbrace 1,2 \rbrace , \lbrace 1,3,4 \rbrace , \lbrace 1,2,3,4 \rbrace , \lbrace 1,2,5 \rbrace \rbrace$.
For $\tau_{M}$ we take subspace topology on $M = \lbrace 1,3,5 \rbrace$.
We consider th... | You have $\tau_{M} = \lbrace \lbrace 1,3,5 \rbrace , \emptyset , \lbrace 1 \rbrace \rbrace$. You missed $\{1,3\}$ and $\{1,5\}$.
$(f \mid_{M})^{-1}(0) = \{1,3\}$ an open set & $(f \mid_{M})^{-1}(1) = \{5\}$ not an open set. So $f \mid_{M}$ is not continuous.
| {
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"timestamp": "2023-03-29T00:00:00",
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The minimum number of variables to represent a 3D line in a unique way To my best understanding, the minimum number of variables to represent a line in 3D space is four. It means you need at least four values to identify a 3D line. For example from here
$$a x + b y + c z = d \tag{1}$$
defines a line with four variable... | Maybe $5$? $x,y,z$ of the reference point and a unit vector, which needs $\theta$ and $\phi$?
| {
"language": "en",
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"source": "stackexchange",
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mathematical induction natural number Tell me about this exercise, I try to solve it but it was confusing
A bank gives 20\$ and 50\$. I must use mathematical induction so that the bank will create whatever amount of money bigger or equal to 40\$, that it is multiple to 10.
Prove that for every natural number $n≥4$ ther... | As you observed, all we need to show is that for each natural number $k\geq4$, there exists nonnegative integers $m,n$ so that $k=2m+5n$. First, note that if $k=2m+5n$, then $k+2=2(m+1)+5n$, hence if $k$ can be written in desired form, so can $k+2$. But obviously $k=4,5$ work. So by induction, all other integers work t... | {
"language": "en",
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"source": "stackexchange",
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Find g(Z) and show that is holomorphic Let be $\gamma$ the circumference of the center at the origin and radius 2, and we will consider the function $g:\mathbb{C}/\gamma \to\mathbb{C}$
$g(z)=\displaystyle\int_{\gamma}\frac{cos(s)}{z-is}ds$
I have doubts about how to calculate $g(z)$ for $ \left |z\right|\ne2 $ and then... | Define $h(z)=g(iz)$. Then $h$ is holomorphic with derivative $h'(z)=-1/(2\pi )\oint_{\gamma}\cos s/(z-s)^2\operatorname ds$, by Cauchy's differentiation formula.
But then $g'(z)=-ih'(-iz)$.
Meanwhile, for $|z|\gt2$, we have $g(z)=0$, by Cauchy's theorem.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Variance identity for i.i.d mean-zero random vector. Let $Z_1, \ldots, Z_k$ independent mean-zero random vectors in $\mathbb{R}^n$. Show that
\begin{equation}
\mathbb{E} \left\| \sum_{j=1}^k Z_j \right\|_2^2 =
\sum_{j=1}^k \mathbb{E} \left\| Z_j \right\|_2^2
\end{equation}
Answer in correct place. Thank you all for th... | As we know:
\begin{equation}
\left\| Z \right\|_2^2 = \langle Z, Z \rangle = Z^T Z = \sum_{i=1}^n z_i z_i
\end{equation}
with $z_i$ being the $i$-th component of $Z$ vector. We can rewrite the left hand side as
\begin{align}
\mathbb{E} \left\| \sum_{j=1}^k Z_j \right\|_2^2 &= \mathbb{E} \left\langle \sum_{j=1}^k Z_j, ... | {
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Finding a special prime The prime number $$p=82\ 954\ 517$$ has the property that the numbers $$2!+p,3!+p,\cdots , 11!+p$$ are all prime, but $12!+p$ is composite.
Upto $10^{10}$, the only other prime with this property is $105\ 204\ 557$
Does a prime $p$ exist such that $$2!+p,3!+p,\cdots , 12!+p$$ are all prime ? If... | Just a few restrictions:
*
*first leads to p is 5 mod 6
*second eliminates 29 mod 30
*third eliminates 11 mod 30
*fourth eliminates 6 mod 7 ( aka 167,83 mod 210)
*fifth eliminates 1 mod 7 ( aka 197, 113 mod 210)
*sixth eliminates 9 mod 11
*seventh eliminates 6 mod 11
*eighth eliminates 10 mod 11
*ninth elim... | {
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"timestamp": "2023-03-29T00:00:00",
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Evaluate $\lim_{n \to \infty} \sqrt[n^2]{2^n+4^{n^2}}$ Evaluate $$\lim_{n \to \infty} \sqrt[n^2]{2^n+4^{n^2}}$$
We know that as $n\to \infty$ we have $2^n<<2^{2n^2}$ and therefore the limit is $4$
In a more formal way I started with:
$$\log(L)=\lim_{n \to \infty} \log(2^n+4^{n^2})^{\frac{1}{n^2}}=\lim_{n \to \infty}\fr... | Hint:
$n>1$;
$f(n):=4(1+\dfrac{2^n}{2^{2n^2}})^{1/n^2}=$
$4(1+\dfrac{1}{2^{2n^2-n}})^{1/n^2}$;
$4(1+0)^{1/n^2} \lt f(n) < 4(1+1)^{1/n^2}.$
Take the limit.
Recall:
For $a>1$, real; and $n >1$, integer:
$1<a^{1/n^2} <a^{1/n}$, and
$\lim_{n \rightarrow \infty} a^{1/n}=1.$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find the Fourier coefficients of $g$ Suppose $f:\mathbb{R}\rightarrow\mathbb{R}$ is periodic with period $2\pi$ so that $\hat{f}(n)=\frac{1}{1+n^{2}}$ for every $n\in \mathbb{N}$, and $g:\mathbb{R}\rightarrow\mathbb{R}$ is periodic with period $2\pi$ and defined by the formula $g(x)=\int_{0}^{x}f(t)dt $ for every $-\pi... | Since you consider $\hat g (n) $, I'll assume that it's $\mathbb Z$ instead of $\mathbb N$ and that you are using the exponential.
If $f$ is good enough (continuous, for instance; we need this to be able to exchange the integrals), you can write
\begin{align}
\hat g(n)&=\frac1{2\pi}\,\int_0^{2\pi} g(x)\,e^{inx}\,dx
=\... | {
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compactness in $\ell^p$ space Choose $1 \leq p \leq \infty$, and let $D=\left\{x \in \ell^{p}:\|x\|_{p} \leq 1\right\}$ be a closed ball in $\ell^p$. Try to show that $D \text { is not a compact subset of } \ell^{p}$.
So far I've proved that the sequence of standard basis vectors $\left\{\delta_{n}\right\}_{n \in \math... | Firstly, using F. Riesz's lemma, you can prove that the closed unit ball is compact in a normed space if and only if the respective normed space is finite-dimensional, which $\ell^p$ is not.
Secondly, in a metric space, compactness is equivalent with sequential compactness, so yes, if you find a sequence that has no co... | {
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Proving a set of points of continuity has only irrational elements. Part a)
Prove that for any function $f:\mathbb{R} \rightarrow \mathbb{R}$, the set $C_f$ of points of continuity of $f$
$$
C_f = \left\{ a \in \mathbb{R}: \forall \epsilon > 0, \exists \delta > 0 \forall x,y \left( |x-a| < \delta \text{ and } |y-a|<... | For a), take $\epsilon = \frac{1}{n}$ and define $C_f^n=\{ a \in \mathbb{R} \mid \exists \delta>0, a - \delta < x, y < a + \delta \implies |f(x)-f(y)|< \frac{1}{n}\}$. We have that $C_f^n$ is open because given $a \in C_f^n$ and $a - \delta < b < a + \delta$, we can take $\delta_1=\frac{1}{2}\min(a + \delta - b, b - a ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Find the set of values of x for which $\lvert \frac {x-1}{x+1} \rvert <2 $ How to solve this inequality question involving modulus?
I can’t get the same answer as the book [answer below]
I know the properties of absolute values, that is
If $\lvert x \rvert <k$ , then $-k < x < k$.
So for this question, this is my worki... | The problem with the analysis of the second part is that you have taken $|\frac{x-1}{x+1}|
= \frac{x-1}{x+1}$ and done the analysis. This is false : what about those $x$ for which we have $|\frac{x-1}{x+1}| = \boxed{-\frac{x-1}{x+1}}$?
You need to break your analysis according to where $|\frac{x-1}{x+1}| = \frac{x-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3519520",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the law of a random variable
Let $X$ be a discrete random value taking values in $\mathbb{N}^* = \left\{1, 2, 3, \ldots \right\}$, and such that $\exists p \in (0,1) \forall n \geq 1$:
$$
\mathbb{P}[X=n] = p \mathbb{P}[X \geq n]
$$
Find the law of $X$.
I understood the definition of law of random variable b... | You can build a simple recursion for $p_n := P(X=n)$ as follows:
*
*$p_1 = p\underbrace{\sum_{n\geq 1}p_n}_{=1} = p$
*$p_n - p_{n+1} = p\left(\sum_{k\geq n}p_k - \sum_{k\geq n+1}p_k\right) = pp_n\Leftrightarrow p_{n+1} =(1-p)p_n$
All together: $p_1 = p, p_{n+1} = (1-p)p_n \Rightarrow \boxed{p_n = p(1-p)^{n-1}}$
| {
"language": "en",
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"If x - a is a factor of polynomial P(x), then a is a factor of the constant term of the polynomial." - Confused with proof I have recently started learning about polynomials. I've been able to grasp polynomial long division algorithm and the remainder and factor theorems and also a few other common-sense theorems abou... | This does not answer your very question, but the theorem seems easy.
If $(x-a)$ divides $P(x)$ then $P(a)=0$. But $P(a)=a(p_na^{n-1}+\cdots+a p_2+p_1)+p_0$ and $p_0=-a(\cdots)$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Given a Möbius Transformation $w=f(z)$ find $f(-\bar{z})$ Let $w=f(z)$ be a Möbius Transformation, and let $\gamma,\Gamma\subset\mathbb{C}$ be the following curves:
$$\gamma=\{z\in\mathbb{C}\mid \Re(z)=0\}\\ \Gamma=\{w\in\mathbb{C}\mid |w-w_0|=r\}$$
Where $w_0\in\mathbb{C}$ and $r\in\mathbb{R^+}$. Given that $f(\gamm... | $z$ and $-\bar z$ are symmetric about the line $\gamma$. Such points are conjugate.
Mobius transformations happen to preserve conjugate points. Thus $h(z)$ is symmetric to $f(z)$ relative to the circle $\Gamma$. This means they are the images of each other under circle inversion.
So we get $h(z)=w_0+\dfrac{r^2(
w-... | {
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Let $f:[-1,1]\rightarrow \Bbb R$ and $f(\sin(\frac{1}{n}))=\cos(\frac{1}{n})$ and $f'(0)$ exist. Prove that $f(0)=1$
Let $f:[-1,1]\rightarrow \mathbb{R}$ and $f(\sin(\frac{1}{n}))=\cos(\frac{1}{n})$
and $f'(0)$ exist. Prove that $f(0)=1$.
Wwhat I did is because $f'(0)$ exist then $f$ is continuous at $0$ and
$f(0)... | Perfect answer! Your arguments are correct.
| {
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"timestamp": "2023-03-29T00:00:00",
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What does it mean to prove if $p$ then $q$? Does that mean to prove $p\rightarrow q$ is a true statement? Then since when $p$ is false, $p\rightarrow q$ is vacuously true, do I only have to prove $q$ is true when $p$ is true?
| That is the gist of it, yeah.
There are, in practice, several ways to do this, and here is a short summary. A direct proof uses intermediate, already-known implications chained together like this.
$$
p\to p_1\\
p_1\to p_2\\
\vdots\\
p_n\to q
$$
A contrapositive proof is a direct proof of the statement
$$
\text{not }q\t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Can any polynomials in the rational field be decomposed like this I' ve learned that the following examples can be used to decompose a
factor in this way:
x^5 - 5 x + 12 = (x - a) (a^4 - (5 a^3)/8 + (7 a^2)/8 + 1/8 (5 a^2 - 12 a) +
1/8 (5 a^3 - 12 a^2) + ((5 a^4)/16 - a^3/8 + (7 a^2)/16 -
3/16 (5 a^2 - 12 a... | It seems to me that you are asking two separate questions. One is whether every $f$ can be factored as $(x-\alpha)g(x)$ where the coefficients of $g$ are simple expressions in (rationals and) $\alpha$. The other is whether every $f$ can be factored as $(x-\alpha)(x-p_1(\alpha))\cdots(x-p_{n-1}(\alpha))$ where each $p_i... | {
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Is $G= \mathbb{Z}$ a group with binary operation defined as $a \cdot b \equiv a - b$?
Is $G= \mathbb{Z}$ a group with binary operation defined as $a \cdot b \equiv a - b$?
I am pretty sure the answer is NO but I would like some verification on my reasoning below:
Associativity fails as in:
Let $a,b,c \in G$ then $a... | You don't need to disprove every property for $(\mathbb Z,-)$ not to be a group, it suffices to show that at least one property does not hold.
Indeed, as you rightly pointed out, associativity does not work, so it is not a group.
If I tell you that every cat has whiskers, a tail and four paws, it suffices for you to sh... | {
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Field Extension of $\mathbb{R}\left(x+\frac{1}{x}\right)$ Can someone help me to prove $[\mathbb{R}(x):\mathbb{R}\left(x+\frac{1}{x}\right)]=2$? My intuition the basis is $\{x,x+\frac{1}{x}\}$ but I cannot prove. Thank you
| The minimal polynomial of $x$ is $t^2-(x+1/x)t+1$.
To see that $x$ doesn’t lie in $\Bbb R(x+1/x)$, assume first that it does. $x =P(x+1/x)$ where $P$ is a polynomial.
$x= x^{-n}+a_{-n+1}x^{-n+1}+...+a_{n-1}x^{n-1}+x^n$, so $x^{-n}+a_{-n+1}x^{-n+1}+...+(a_1 - 1)x+...+a_{n-1}x^{n-1}+x^n = 0$. Therefore $a_1=1$ and all ... | {
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How to solve homogeneous linear recurrence relations with constant coefficients? Consider a sequence $(a_n)_{n\in\mathbb N}$ defined by $k$ initial values $(a_1,\dots,a_k)$ and
$$a_{n+k}=c_{k-1}a_{n+k-1}+\dots+c_0a_n$$
for all $n\in\mathbb N$.
What are some ways to get closed forms for $a_n$? What are some ways of rewr... | Characteristic/Auxiliary Polynomials
The basic solution
*
*Suppose that $\alpha$ is a root of the associated polynomial $$x^k=c_{k-1}x^{k-1}+c_{k-2}x^{k-2}+\dots+c_0\quad(1)$$ Then it is also true that $$\alpha^{n+k}=c_1\alpha^{n+k-1}+\dots+c_0\alpha^n$$ So $a_n=\alpha^n$ satisfies the recurrence relation (but proba... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Reduction from Graph Isomorphism to String Isomorphism I was studying the reduction from Graph Isomorphism (GI) to String Isomorphism (SI) showed in this bachelor's thesis in Chapter 2.2 and was understanding the procedures just fine until I got stuck in a proof.
My problem is in the following Lemma:
Lemma 2.9. If SI ... | In the string isomorphism problem, we are given two length-$N$ strings over the same alphabet and a group $G$ of allowable permutations: a subgroup of $S_N$. The problem is to find all the elements of $G$ which permute one string into the other (or, in the decision version of the problem, to determine if there is any s... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finite harmonic sum inequality I wish to prove some inequality involving a finite harmonic series:
$$\sum_{k=n+1}^{n^2}\frac{1}{k}>\sum_{k=2}^{n}\frac{1}{k}$$
Certainly $\frac{1}{nk+q}≥\frac{1}{n(k+1)}$
for $q=1,2,3,....,n.$
So that $$\sum_{q=1}^n\frac{1}{nk+q}≥\frac{1}{k+1}$$
Adding the last inequality from $k=1$ to ... | You want to prove that
$$ H_{n^2}-H_{n} \geq H_n -1 $$
i.e. that
$$ H_{n^2}-2H_n\geq -1. $$
On the interval $\left[\frac{1}{2},1\right]$ we have $\frac{x-\log(1+x)}{x^2}\in\left[0.3,0.4\right]$, so
$$ \frac{3}{25}\leq\sum_{k=2}^{n}\frac{1}{k}-\sum_{k=2}^{n}\log\left(1+\frac{1}{k}\right)\leq\frac{2}{5}\sum_{k=2}^{n}\fr... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Asymptotic expansions for $\int_x^\infty e^{-y^3} dy$ I would like to find asymptotic expansions of $F(x) = \int_x^\infty e^{-y^3} dy$ as (a) $x \to 0$ and as (b) $x \to \infty$.
I solved (a) using the expansion for $e^{-y^3}$ around $0$:
$$F(x) = \int_0^\infty e^{-y^3} dy - \int_0^x\sum_{j= 0}^\infty (-1)^j \frac{y^{... | The Integral equals $\frac13 \Gamma(1/3,x^3)$ where $\Gamma(a,z)$ is the incomplete gamma function.
You can look up that the asymptotics around $0$ are $\frac{1}{3}\Gamma \left(\frac{1}{3}\right)-x+\frac{x^4}{4}+O\left(x^5\right)$,
and around $\infty$ is $e^{-x^3+O\left(\left(\frac{1}{x}\right)^6\right)} \left(\frac{1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3521445",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Calculus: Find the limit: $\lim_{h\rightarrow0}\frac{f(x+h)−f(x)}{h}$ given that $f(x)=\sin(2x)$ Find the limit:
$$\lim_{h\rightarrow0}\frac{f(x+h)−f(x)}{h}$$
Given that $f(x)=\sin(2x)$.
Tried many ways, but I kept on getting an indeterminate form. I can't find a way to cancel out terms on the numerator and denominator... | Hint
$$\sin(2x+2h)=\sin 2x\cos 2h+\cos 2x\sin 2h$$
$$\lim_{u\to 0}{\sin u\over u}=1$$
$$\lim_{u\to 0}{\cos u-1\over u}=0$$
| {
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Evaluate $\lim_{n\to \infty}\frac{1}{\sqrt[4]{{n^4}+n+2}}+\cdots+\frac{1}{\sqrt[4]{{n^4}+5n-1}}$ I am new to analysis and I have no clue how to solve this limit. This is an exam problem from my analysis 1 course, there are one or two similar ones on the exam.
$$\lim_{n\to \infty}\frac{1}{\sqrt[4]{{n^4}+n+2}}+\cdots+\fr... | Also note that $\frac{1}{n+1}<\frac{1}{n^4+n+k} <\frac{1}{n}$
so $$ \sum_{k=1}^{4n-1} \frac{1}{n+1}<S_n=\sum_{k=1}^{4n-1} \frac{1}{(n^4+n+k)^{1/4}} <\sum_{k=1}^{4n-1} \frac{1}{n}.$$ So, we note that
$$ \lim_{n \rightarrow \infty} \frac{4n-1}{n+1}=4= \lim_{n \rightarrow \infty} \frac{4n-1}{n}= \lim_{n \rightarrow \inf... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3521700",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Comparing the Dual Banach Space with norm topology and Weak* topology If we consider the dual Banach space $V' :=\{f:V\rightarrow \mathbb{C}$ such that $f$ is linear and bounded$\}$. We know $V'$ also forms a Banach space in the norm topology where the norm is the general operator norm. But the open ball is not compact... | A subbasic open set of the weak$^\ast$ topology is of the form $O(v,\epsilon):=\{f \in V': |f(v)| < \epsilon\}$ (for some $v \in V, \epsilon>0$ and so for every $f \in O(v,\epsilon)$, we have $f \in B_d(f, \epsilon) \subseteq O(v, \epsilon)$, where $d$ is the sup-operator norm and so every weak$^\ast$ open set is opera... | {
"language": "en",
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$O(h^3)$ in the second-order approximation for $f(\mathbf{x}^*)$ I am currently studying the textbook Algorithms for Optimization by Mikel J. Kochenderfer and Tim A. Wheeler. Chapter 1.6.2 Multivariate says the following:
The following conditions are necessary for $\mathbf{x}$ to be at a local minimum of $f$:
*
... | The term $O(h^3)$ means that the estimation error is locally bounded by a third degree polynomial. For example, the second order estimation of $f(x)=e^x$ at $x=2$ is $g(x) = e^2(0.5x^2 - x + 1)$, so $f(x) = g(x) + O(x^3)$. The higher the power of the error term, the more rapidly it goes to $0$ as $x\to 2$. Note that th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3521915",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
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Infinite series: defining the sum Consider the following sum described below:
$$\sum_{i=0}^{x}\frac{1}{\sqrt{(3+i)(2k-4-i)}}$$
Where $0 \leq x \leq 2k-8$ and even and $k\geq 5$ is a constant integer.
I need to find the closed form expression for this sum, however, after many attempts I couldn't. This would make it ea... | To show that it is increasing, consider $$a_i=\frac1 {\sqrt{(3+i)(2k-4-i)}}$$ and let $$b_i=\frac 1 {a_i^2}=(3+i)(2k-4-i)\implies b_{i+1}-b_i=2k-2i-8$$ So $(b_{i+1}-b_i)$ is decreasing with $i$ for a given $k$ and so $(a_{i+1}-a_i)$ is increasing.
For a closed form, I am quite skeptical (even using special functions)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3522101",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Get complex angle when computing quaternions (Satellite attitude) I am modelling a, so far uncontrolled, satellite in MATLAB , along with its attitude. I have been researching about this matter and found that quaternions are the way to go , since they don't have singularities. My goal was to get angular velocities in e... | Here's a reference to how the 3D game engine people do rotations using quaternions: https://www.3dgep.com/understanding-quaternions/
In particular, for this problem, I suggest working through SLERPing. I suspect you might find there is a better modelling solution by staying within quaternion arithmetic.
Enjoy! (It's n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3522318",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Why does this would eventually simplify into the original circle equation? I was trying to solve this problem:
The point $A$ has coordinates $(5, 16)$ and the point $B$ has coordinates $(-4,4)$. The variable $P$ has coordinate $(x,y)$ and moves on a path such that $ AP = 2BP$. Show that the Cartesian equation of the pa... | The locus of the points such that the ratio of their distance to two given points is constant is a circle:
$$\frac{\sqrt{(x-x_1)^2+(y-y_1)^2}}{\sqrt{(x-x_0)^2+(y-y_0)^2}}=\lambda.$$
This is because
$$(x-x_1)^2+(y-y_1)^2-\lambda^2((x-x_0)^2+(y-y_0)^2)=0$$ is the equation of a conic, such that
*
*the coefficients of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3522446",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Why the following description implies the limit is finite? Let $f:[0,T]\rightarrow \mathbb{R}$ $$V(f) = \lim_{\|\Pi\|\rightarrow 0}\sum_{j=0}^{n-1}|f(t_{j+1})-f(t_j)|$$
where $\Pi$ is a partition of $[0,T]$ and define $\|\Pi\|$ as
$$\Pi=\{t_0,t_1,\cdots,t_n\}, \ \ 0=t_0<t_1<\ldots<t_n=T, \ \ \|\Pi\|=\max_i(t_{i+1}-t_i... | What are you asking for the converse of what is sated in the manual. The manual says that if $V(f)<\infty$ the something happens. The stated property is immediate from the definition of limit. It is something like this: if $\lim a_n=l$ and $\epsilon >0$ then there exists $N $ such that $a_n <l+\epsilon$ for $n \geq N$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3522602",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Removing a penny raises mean coin value from 17 to 18. How many nickles? So this is a question that my sister (grade 8) got wrong on a test.
There is a collection of quarters, dimes, nickels, and pennies in a jar. The mean of these coins is 17 but when you remove a penny, the mean becomes 18.
How many nickels are in t... | You got most of the way there already. You know there are 16 coins that are worth a total of 288 cents. Three of these must be pennies (since 288 is 3 mod 5 and all non-pennies have value 0 mod 5), so that leaves 13 worth 285.
This can be achieved with 11 quarters and 2 nickels.
In a jar with $Q$ quarters and $13-Q$ o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3522725",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Show that the sequence $x_n=\sum\limits_{k=1}^n\frac1{\sqrt{k+1}+\sqrt{k}}$ is unbounded. Consider the sequence $\{x_n\}_{n\ge1}$ defined by $$x_n=\sum_{k=1}^n\frac{1}{\sqrt{k+1}+\sqrt{k}}, \forall n\in\mathbb{N}.$$ Is $\{x_n\}_{n\ge 1}$ bounded or unbounded.
I solved the problem as stated in the answer posted by me. ... | $\sqrt {k+1}+\sqrt k \leq \sqrt {2k}+\sqrt k<3\sqrt k$. Hence the given sum is at least $\sum\limits_{k=1}^{n} \frac 1 {3\sqrt k}$ Now use the fact that the series $\sum\limits_{k=1}^{n} \frac 1 {3\sqrt k}$ is divergent.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3522867",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Without use of Darboux's theorem, prove that $f'$, where $f(x)=x^2\sin\left(\frac{1}{x}\right)$, enjoys IVP
Prove (wihout use of Darboux's theorem) that the derivative of the function: $$f(x)=\left \{\begin {array}{ll}
x^2\sin\left(\frac{1}{x}\right)&,~x\neq0\\
0&,~x=0\\
\end{array}
\right.,$$ that is
$$f'(x)=\left ... | One keeps all definitions as in the question.
Lemma. Let $I$ be an interval either of the form $(\alpha ,0]$ or $[0,\beta)$. Let $J:=I\setminus \{0\}$. The $f'(J)$ contains the interval $(-1,1)$.
Proof. It is clear that $$\limsup_{x\rightarrow 0^+}f'(x)=1,\liminf_{x\rightarrow 0^+}f'(x)=-1,$$ and similarly
$$\limsup_{x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3523018",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Is it good to mix up syntax with semantics in logic? just want to know is it "good" to use syntax and semantics together in a formal prove in mathematics.
With Completeness theorm,syntax of first order logic is equal in value to semantics. However,in Zhongwang Lu's mathematical logic towards computer science (the book ... | In many low-level foundations of mathematics introductions there is a lack of precision if it is about completeness and soundness. For example, there is a quite popular "mixing up syntax with semantics"-confusion over the definition of completeness. Some authors use the notion of "completeness" for both:
*
*semantic... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3523162",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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How to find all possible minimal edge covers over $K_6$? How to find all possible minimal edge covers over $K_6$?
$K_6$-complete graph of $6$ vertices.
On working out I found out that there would be these many cases ..
Can somebody tell me if this what I have done is right?
Or are there any more cases?
| OEIS A053530 provides an exponential generating function
$$\exp(-x - x^2/2 + x\exp(x)),$$ and the count for $n=6$ is 171, which confirms that your 111 plus the missing 60 cover all cases.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3523398",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Prove that for any $n > 0$, if $a^n$ is even, then $a$ is even the proof at hand is:
Prove that for any $n > 0$, if $a^n$ is even, then $a$ is even.
Hint: Contradiction
So I know to start the problem in a contradiction format would be:
$a^n$ is even, then $a$ is odd so that $a = 2k+1$. Then plug in that into $a^n$, ... | The contrapositive is that $$a \text{ odd }\implies a^n \text { odd}$$
Use induction here: if $a^1=2k_1+1$, then $a^2=(2k_1+1)^2=2(2k_1^2+2k_1)+1=2k_2+1$
hence true for $n=2$
Then assume true for $n=m$, $(2k_1+1)^m=2k_m+1$
Then $$(2k_1+1)^{m+1}=(2k_m+1)(2k_1+1)=4k_mk_1+2k_m+2k_1+1=2(2k_mk_1+k_m+k_1)+1$$
Hence true with... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3523495",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 3
} |
density of the product of uniformly distributed random variables Let $X$ and $Y$ be independent with uniform distribution over $(0,a)$ and set $Z=X^2Y^2$. What is the joint density of $Z$
\begin{align}
F_{Z}(t) &= \mathbb P(X^2Y^2<t)
= \begin{cases}
0,& t<0\\
1,& t>a^4
\end{cases}
\end{align}
Consider the case when $0... | No. You should integrate below the hyperbola $yx=\sqrt{t}$ or $y=\frac{\sqrt{t}}{a}$ at the picture:
So
$$
F_{Z}(t) = \mathbb P(XY<\sqrt t)=1/a^2 \int_0^{\sqrt t/a}dx\int_0^a dy + 1/a^2 \int_{\sqrt t/a}^a dx\int_0^{\sqrt t/x} dy
$$
Here is slightly modified picture with integration bounds for $y$ are drawn at each ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3523607",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to find out the probability that the tallest person in a group of people is a man? Assume we have a population of N men and women such that exactly $N/2$ people are men (set $M$) and $N/2$ people are women (set $W$).
Assume that the standard deviations for height between both groups is the same $\sigma$ however the... | Let
\begin{align*}
M_1, \cdots, M_{N/2} &\overset{\text{iid}}{\sim} N(\mu_M, \sigma^2) \\
W_2, \cdots, W_{N/2} &\overset{\text{iid}}{\sim} N(\mu_W, \sigma^2) \\
S_M, S_W &\overset{\text{iid}}{\sim} \text{SRSWOR}(\{1, \cdots, N/2\}) \\
\end{align*}
where SRSWOR meaning simple random sample without replacement. Define
\b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3523710",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
Measurability of a stopped local martingale Let $X_t$ be a local martingale.
Let $\tau_n$ be a localizing sequence.
We then know that $$E[X_{t\land \tau_n}|\mathcal F_s]=X_{s\land\tau_n}$$
$1)$ But $X_{s\land\tau_n}$ is measurable w.r.t wich sigma algebra? is it $\mathcal F_s$ measurable?
$2)$ Moreover, if i have $E[e... | Yes. It is measurable w.r.t $\mathcal F_{s \wedge \tau_n}$ and $F_{s \wedge \tau_n} \subset \mathcal F_s$.
$e^{t\wedge \tau_n}$ is measurable w.r.t. $\mathcal F_t$. So it is legitimate to bring it to RHS only when you know that $t \leq s$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3523804",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove that if $a\nmid b$, $ax^3+bx+(b+a)=0$ has no natural number solutions Let $a,b\in\mathbb Z$ with $a\neq0$. I need to prove that if $a\nmid b$, then the equation $ax^3+bx+(b+a)=0$ does not have a solution that is a natural number.
I noticed that regardless of the values of $a$ and $b$, the equation will always hav... | Your quadratic equation is, with the factor of $a$ included,
$$ax^2 - ax + a + b = 0 \implies b = -a(x^2 - x + 1) \tag{1}\label{eq1A}$$
As Bill Dubuque's question comment states, for $x \in \mathbb{N}$, you have $a$ dividing the right hand side, so it must also divide the left hand side, i.e., $b$. Thus, you require $a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3523960",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How do we use the Intermediate Value Theorem to show that $2^x = \frac{10}{x}$ for $x>0$. The Intermediate Value Theorem states that over a closed interval $[a,b]$ for line $L$, that there exists a value $c$ in that interval such that $f(c) = L$.
We know both functions require $x>0$, however this is not a closed inte... | You have an error in one of the steps it should be as follows:
$$x2^x=10 \implies \ln(x2^x)=\ln 10 \implies \ln x+ x \ln 2=\ln 10.$$
To use IVT, let $f(x)=x2^x-10$, now argue that it is a continuous function for $x>0$. After this find two positive real numbers $a$ and $b$ such that $f(a)<0$ and $f(b)>0$. Then by IVT yo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3524118",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Show that $f(z) = \text{Im }z$ is not differentiable I want to prove that $f(z) = \text{Im }z$ is not differentiable anywhere. I know how to prove it easily with the Cauchy-Riemann equations, however I'm also interested in proving it by just using the definition of differentiability.
I know that the definition of diff... | For any $z\in \mathbb{C}$ consider the limit of the difference quotient in two different directions, namely in the purely imaginary direction and also in the real direction. These limits will not agree, so $\text{Im }z$ is not complex differentiable.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3524362",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
How can we count different coloring in 2 cycle with one edge in common There is a question that asked what is the counts of different coloring in $C_5$ and $C_6$ when it has one edge in common:
I have used fundamental reduction in cycles in order to get a recursive formula (I know there is a closed formula for that to... | Let's first find a formula for a "mouse graph" ($C_n$ with a path $P_m$ attached)
This is easy with induction and the fundamental d-c. We know that for $n$-cycle the chromatic polynomial is $C_n(x) = (x-1)^n+(-1)^n(x-1)$ and we get that for the mouse it is
$$M_{n,m}(x) = (x-1)^m C_n(x)$$
Now for the graph in question.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3524470",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Find all polynomials $P$ for which $(P(x)-x)\mid P^{(n)}(x)-x$ $n\gt1$ is a fixed natural number. Find all polynomials $P(x)$ with complex coefficients for which $(P(x)-x)\mid P^{(n)}(x)-x,$ where $P^{(n)}()$ is the $n$th iterate: $P^{(1)}(x)=P(x)$ and $P^{(i+1)}(x)$ = $P(P^{(i)}(x))$
.
What I proved until now : I pro... | If $r$ is a root of $P(x)-x$ of order $m$, i.e. $P(x) = x + O((x-r)^m)$ as $x \to r$, then
I claim $P^{(n)}(x) = x + O((x-r)^m)$ as well. This should be possible to prove by induction on $n$. Therefore all roots of $P(x) - x$ are roots of $P^{(n)}(x) - x$ with the same or greater multiplicity. We conclude that $P(x)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3524594",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
$e^{-xy} + e^{xy} = 2e^{-y}$ - where am I going wrong? I am trying to see if there is any $x$ (real or complex) for which this equation can be solved.
$$e^{-xy} + e^{xy} = 2e^{-y}$$
Step 1. Multiplying both sides by y,
$$ye^{-xy} + ye^{xy} = 2ye^{-y}$$
Step2. Partially differentiating the original equation (i.e. $e^{-x... | The derivative of two different constant functions is zero. The two functions are not equal. All equality of derivatives tells you is that the two functions agree up to a constant offset (recall the "${}+C$" from integral calculus).
The partial derivative with respect to $x$ of two different functions depending only ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3524722",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
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