Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Power series such that the $n$-th partial sum has $n$ distinct roots I am looking for examples of power series of the form
$$\sum_{k=0}^\infty a_k x^k$$
(where $a_k \in \mathbb{C}$ for all $k$) such that the polynomial given by its $n$-th partial sum has $n$ distinct roots, i.e.:
$$\sum_{k=0}^n a_k x^k$$
has $n$ distin... | Let $a_k$ be a sequence of numbers that are algebraically independent over the rationals. Thus for any nontrivial polynomial $p(x_0, \ldots, x_n)$ with rational coefficients, $p(a_0, \ldots, a_n) \ne 0$.
The polynomial $P_n(x) = a_0 + a_1 x + \ldots + a_n x^n$ has a repeated root if and only if its discriminant is $0$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3294332",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
On Composite Numbers of the Form $p_{1}p_{2} \ldots p_{k} - 1$ This question is related to D. H. Lehmer's 1932 conjecture on Euler's totient function: Are there any composite $n$ for which $\phi(n)$ divides $n-1$?
See, for example:
On Lehmer's Totient Conjecture
I would like to ask what is known regarding the factors ... | Here are some additional things related Lehmer's totient conjecture that is known.
Definition: If $n$ is composite then $\phi(n)<n−1$, hence there is at least one divisor $d$ of $n−1$ which does not divide $\phi(n)$. We call $d$ as the totient divisor of $n$. Trivially, if $n$ is prime then it has no totient divisor an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3294459",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Why does a non square matrix lack a multiplicative identity Precisely, why is multiplicative identity defined to be $IA=AI=A$ why both sides should work why not something like $AI=A$ ? Is there an underlying advantage?
| The use of category theory may make this clearer. Suppose we have a category of objects
$\, V_1, V_2, V_3, \dots\,$ An $\,n \times m\,$ matrix $\,A\,$ is an arrow from
$\,V_m\,$ to $\, V_n.\,$ Matrix multiplication is only defined between compatible matrices.
That is, If $\,B\,$ is an arrow from $\,V_n\,$ to $\,V_k\,$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3294572",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
$\lim_{x \rightarrow \infty} \frac{f(x)}{g(x)}$ where $f(x)$ is linear and $g(x)$ is either strictly convex or strictly concave I am currently working on a problem where I am interested in the limit of a ratio of functions,
$$\lim_{x \rightarrow \infty} \frac{f(x)}{g(x)}.$$
It is known that $f(x)$ is linearily increasi... | The limit can be different from $0$. Actually it can be any real number $m$. Consider for example $f(x)=mx$ and $g(x)=x-\arctan(x)$ which is strictly convex for $x>0$ (or $g(x)=x+\arctan(x)$ for a strictly concave one). Then
$$\lim_{x \rightarrow +\infty} \frac{f(x)}{g(x)}=m.$$
It can be also $\pm \infty$. Take $f(x)=\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3294687",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
If $x,y,z\in\mathbb{R}^+$, prove that $\sqrt{x^2-xz+z^2}+\sqrt{y^2-yz+z^2}\ge\sqrt{x^2+xy+y^2}.$ When I was doing Math Training, the coach gave a inequality problem.
If $x,y,z\in\mathbb{R}^+$, prove that $$\sqrt{x^2-xz+z^2}+\sqrt{y^2-yz+z^2}\ge\sqrt{x^2+xy+y^2}.$$
I tried to use the brute force method, and found out ... | Hint: After one times squaring we get
$$2\sqrt{x^2-xz+z^2}\sqrt{y^2-yz+z^2}\geq xz+yz+xy-2z^2$$
Squaring again and factorizing we obtain
$$(xy-xz-yz)^2\geq 0$$ which is true.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3294805",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Not understanding the definition of a differential of a map. I'am reading Loring.W.Tu's book on Manifolds and I'am stuck at the point where he defines differential of a smooth map between smooth manifolds $N$ and $M$.
If $F:N\rightarrow M$ is a $C^{\infty}$ map between two manifolds (smooth) then at each point $p\in N$... | They talk about this in example 8.4. Let $f:\mathbb{R}^n\rightarrow\mathbb{R}^m$ be smooth and $p\in \mathbb{R}^n$. Take standard coordinates of $\mathbb{R}^n$ and $\mathbb{R}^m$ via $(x^1,\cdots, x^n)$ and $(y^1\cdots, y^m)$, respectively. Then, the map $F_*:T_p\mathbb{R}^n\rightarrow T_{F(p)}\mathbb{R}^m$ is a linear... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3294959",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Associated primes in a reduced ring
Let $R$ be a reduced ring. Show that $\operatorname{Ass}R$ is the set of minimal prime ideals of $R$.
I think that the first inclusion must come from using $\operatorname{Ass}R \subseteq\operatorname{Supp}R$, assuming the ideal is not minimal, and then showing some contradiction. N... | If $R$ is reduced and $P=\operatorname{Ann}(x)$ is an associated prime, suppose $Q$ is a prime properly contained in $P$. Then $(x)P\subseteq Q$ implies $x\in Q$. But then $x^2=0$, a contradiction. So $P$ was already minimal.
The other direction isn't clear to me. In this paper they talk about necessary and sufficien... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3295064",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How to define measure over infinite sequence of random variables Suppose we are drawing elements from a set $\Omega$ at each time $t\in \mathbb N$. The probability of drawing $\omega\in \Omega$ at time $t$ is given by a conditional distribution $\mu(\omega|\omega_1,\omega_2,\ldots,\omega_{t-1})$ (the function $(\omega_... | The Kolmogorov Extension Theorem may be able to help. The "Implications of the Theorem" section seems relevant to your question here. It's not constructive at the infinite level, but provides a formalism to link the finite to the infinite.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3295172",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Roots of $\sum_{k=0}^n x^k$ How can I go on to show that the roots of $1+x+x^2+x^3+\ldots+x^n$ are exactly
$$\exp\left(\frac{2ki\pi}{n+1}\right)$$
for $k=1,\ldots,n$?
| $(1-x)(1+x+x^{2}+...+x^{n})= 1-x^{n+1}$. So the roots are same as the roots of $1-x^{n+1}=0$ except for $x=1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3295282",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Proving that limits are a "local property" From M. Spivak's Calculus, let me reproduce Problem 10 in the chapter Limits.
Suppose there is a $\delta>0$ such that $f(x) = g(x)$ when $0<|x-a|<\delta$. Prove that $\lim_{x\to a}f(x)=\lim_{x\to a}g(x)$. In other words, $\lim_{x\to a}f(x)$ depends only on the values of $f(x)... | What you have shown is the following:
If $\lim \limits_{x \to a}f(x)$ and $\lim \limits_{x \to a}g(x)$ exist, and there is a $\delta'>0$ such that $f(x) = g(x)$ whenever $0<|x-a|<\delta'$, then $\lim \limits_{x \to a}f(x) = \lim \limits_{x \to a}g(x)$.
Your proof of this assertion seems fine.
As suggested in the comm... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3295680",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How can I plot $f(x, y) = x^2 + y^2$ I want to plot $f(x, y) = x^2 + y^2$?
I can plot functions of a single variable but I don't know how to plot multivariable function.
| The graph of this function will be a surface in space. Above the point $(x,y)$ in the plane it has height $f(x,y)$.
https://www.mathcurve.com/surfaces.gb/paraboloidrevolution/paraboloidrevolution.shtml
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3295839",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Is this substitution correct?
I think the second equation above is incorrect. It seems to me that the constant k will cancel out. Am I interpreting the equation for the price elasticity of demand incorrectly as equivalent to:
$E = \frac{\partial q}{\partial p} * \frac{p}{q} $
| Warning: I won't make the $t$ subscripts explicit.
The source you quote seems to have a misprint. I'm no economist, but$$E=\frac{p}{q}\frac{\partial q}{\partial p}\implies\frac{\partial\ln y}{\partial p}=\frac{1}{q}\frac{\partial q}{\partial p}=\frac{E}{p}.$$For constant $E$, this integrates to $\ln y=E\ln p+\text{cons... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3295986",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Show that composition by $\varphi$ is a linear mapping
Let $\varphi$ be any mapping from a set $A$ to a set $B$. Show that composition by $\varphi$ is a linear mapping from $\mathbb{R}^B$ to $\mathbb{R}^A$. That is, show that $T:\mathbb{R}^B \rightarrow \mathbb{R}^A$ defined by $T(f) = f \circ \varphi$ is linear.
To... | $T$ is a function on $\mathbb{R}^B$, so you need to show it is linear on this space. That is, if $c\in\mathbb{R},$ and $f,g\in \mathbb{R}^B,$ then you need to show that $$T(cf)=cT(f),$$ and $$T(f+g)=T(f)+T(g).$$ You just need to use composition properties to show these; I'm sure you can take it from here.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3296083",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Notions of continuity for stochastic processes I would like to receive some clarification regarding the difference between continuous in probability and continuous almost surely. Using the definition of the wikipedia page (that match the one I have seen in other references), we have
Continuous in probability
for all $\... | 1. We first examine a much easier variant, i.e., discrete-time process. Let $Y = (Y_n)_{n\in\mathbb{N}_1}$ be a sequence of independent random variables such that
$$ \mathbb{P}(Y_n = 1) = \frac{1}{n} \qquad\text{and} \qquad \mathbb{P}(Y_n = 0) = 1 - \frac{1}{n}. $$
Then it is clear that, for each $\epsilon > 0$,
$$ \li... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3296182",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Characteristic Direct product of the Rings Let the rings, $R_i$ and its direct product, $R = \Pi_1 ^{n}R_i = R_1 \times R_2 \times ... \times R_n$
Say the $Char(R_i) = m_i$
(1) $Char(R)$ = $lcm(m_1,m_2,...m_n)$
If all the rings, $R_i$ are commutative, The statement (1) is surely true.
But what if the There are some ri... | Generally, if you have two groups $G_1$ and $G_2$, then the order of an element $ (g_1, g_2) \in G_1 \times G_2$ is simply lcm(ord($g_1$), ord($g_2$)). If the order of one of the elements is infinite, we simply take the lcm to mean infinite.
The underlying additive group of any ring is abelian. And the characteristic ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3296368",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
$x,y,z$ are all strictly positive, $x+y+z=1$, what is $\max(xyz)$ $x,y,z$ are all strictly positive, $x+y+z=1$, what is $\max(xyz)$?
My attempt:
Using rand() function in Microsoft Excel to generate random numbers between $0$ and $1$. I used this function for the values of $x$ and $y$.
For the value of $z$, I used the f... | There are three ways that I can come up with.
*
*By using the AM-GM inequality, you have
$$\sqrt [3]{xyz} \leq \frac{x+y+z}{3},$$
for non-negative $x$, $y$, and $z$.
*As you attempted, represent $z$ with $x$ and $y$, and compute partial derivatives of $xyz$ with $z$ being replaced by $1-x-y$, and check if the cr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3296475",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Find a smooth function $\eta:\mathbb{C}\to\mathbb{R}$ whose support is a disk. This comes from the proof of the following lemma in Jost's Compact Riemann Surfaces (Lemma 2.3.3).
Lemma 2.3.3 Every compact Riemann surface $\Sigma$ admits a conformal Riemann metric.
proof. ... For a disk $D\subset\mathbb C$ we choose a s... | $f(x)=e^{-\frac 1 {1-x}}$ for $x<1$ and $0$ for $x \geq 1$ defines a smooth function which is positive on $(-\infty,1)$ and $0$ outside it. So $f(\|x\|^{2})$ is a smooth function on $\mathbb R^{2}$ which is positive for $\|x\|<1$ and $0$ elsewhere. For any other disk in $\mathbb R^{2}$ use an appropriate affine tran... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3296624",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Limit of derivative function at infinity
$f$ is a differentiable function on real line such that $$\lim_{x \to \infty}f(x)=1$$ and $$\lim_{x \to \infty}f'(x)=\alpha$$. Then what can be said about $\alpha$
*
*$\alpha=0$
*$\alpha$ may not be $0$ but $|\alpha| \le1$
*$\alpha\geq1$
*$\alpha\leq-1$
I cannot think of ... | Because $\lim_{x \to \infty}f(x)=1$, $y=1$ is a horizontal asymptote of the graph of $f(x)$.
Therefore, $\lim_{x \to \infty}f'(x)=0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3296739",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
If $a\leq b$ and $-a\leq b$, then $|a|\leq b$. I have arrived at the two separate conclusions:
*
*$a\leq b$
*$-a\leq b$
Can I conclude that $|a|\leq b$? I am missing something as it is not by definition of the absolute value.
| You could prove it by proving the contrapositive, as follows.
Assume $|a|>b$.
Now $|a|=a$ or $|a|=-a$.
Therefore $a>b$ or $-a>b$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3296820",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Linear regression without intercept: formula for slope To do linear regression there is good answer from TecHunter
Slope;
$$\alpha = {n\sum(xy) - \sum x \sum y \over n\sum x^2 - (\sum x)^2}$$
Offset:
$$\beta = {\sum y - \alpha \sum x \over n}$$
Trendline formula:
$$y = \alpha x + \beta $$
However, How does these formul... | Since $\beta=0$, $\alpha = \frac{\sum y}{\sum x}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3297060",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Optimizing quadratic form with respect to inner positive definite matrix with a trace constraint Let $\{z_i\}_{i=1}^n$ and $\{w_i\}_{i=1}^n$ be two collections of vectors in $\mathbb R^p$. Let $A$ be a real positive definite $p\times p$ matrix, with Cholesky factorization $LL^T$, where $L$ is also $p\times p$.
I want t... | Define the $p\times n$ matrices $Z=[z_1,\dots,z_n]$ and $W=[w_1,\dots,w_n]$ (such that given vectors are respectively their columns). Convince yourself that you can rewrite your optimization problem as
\begin{align}
\min_{A} &<A,ZZ^T-WW^T> \\ &A\geq 0 ~,~<A,I> = 1
\end{align}
where $A\geq 0$ implies $A$ should be posit... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3297190",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
} |
Sum of a Sum of a Squared Difference
How did the author jump from the second equation to the third equation? I suspect there’s a rule I’m forgetting that allows for this, any help is appreciated.
| Note that
\begin{align}
-\left(\sum_{i=1}^{160} (x_i - 8)^2 - \sum_{i=1}^{160} (x_i - 7)^2\right) & =
-\left(\sum_{i=1}^{160} (x_i^2 - 16x_i + 64) - (x_i^2 - 14x_i + 49)\right) \\
& = -\left(\sum_{i=1}^{160} (-2x_i + 15)\right) \\
& = 2\sum_{i=1}^{160} x_i - 2400 \tag{1}\label{eq1}
\end{align}
As you can see, the $15$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3297316",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
A Question on Cardinality $\aleph_{0}$ I'm trying to understand the concept of Cardinality.
My question is,
Let the interval $[1, 2n]$ is given.
In this interval we have $2n$ natural numbers. Or $n\to\infty$, we have countable infinite natural numbers and Cardinality equal to $\aleph_0$.
Then, in this interval we have... | First of all, you try to work with limits and want to use that an expression at the limit point equals the limit of said expression as we approach the limit point. But for that you first of all need to know that the function you consider is defined at the limit point. So, how do you define division at infinity? And eve... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3297414",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
How to derive derivative of the logarithm of a summation? I'm currently reading the book Deep Learning (Goodfellow et al., 2015) and had a question regarding the calculation of a gradient when explaining backpropagation for a certain example. For anyone who's curious, this is from section 6.5.9: Differentiation outside... | Your derivation of $p_i\log q_i$ is fine. Based upon it we obtain for $J$:
\begin{align*}
J&=-\sum_{j=1}^np_jz_j+\sum_{j=1}^np_j\log\left(\sum_{k=1}^ne^{z_k}\right)\\
&=-\sum_{j=1}^np_jz_j+\log\left(\sum_{k=1}^ne^{z_k}\right)\tag{1}
\end{align*}
In the last line we use the sum of the probabilities $p_j,1\leq j\leq n$ i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3297525",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Numerically stable evaluation of $x^{n!}$ Given that $x$ is a real number with property $0 < x < 1$ and $n$ upto $4000$
Is there a good way to decompose the n! into steps for multiplying x?
| Minimal number of multiplications in worst case $n=4000$: $N_{\min}=\lceil\log_2 4000!\rceil=42\,100$. “Bruteforce” approach, using fast multiplication for power 2, then 3, then 4 etc gives $N_{bf}=\sum_{k=2}^{4000} \lceil\log_2k\rceil=43\,905$. So it's less than 5% inoptimal. I wouldn't be bothered with writing a soph... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3297617",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
A question on counting I'm having difficulty answering (Qb iii)
Question:
(a) Write an expression for the number of sets S which contain 10 elements, each of which is
an integer between 1 and 20. | There are ${10 \choose 2}$ subsets of size 2.
(iii) For a non-empty subset X ⊆ S, let t(X) denote the sum of the members of X. Prove
that there must be distinct subsets A,B ⊆ S, each of size two, such that t(A) = t(B).
(Hint: what are the possible values of t(A) and t(B)?)
Where I am at so far
I realised the following:... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3297723",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 2
} |
Find the constant $k$ from the determinant
Given: $$\begin{vmatrix}(b+c)^2 &a^2&a^2\\b^2 &(c+a)^2&b^2 \\c^2&c^2& (a+b)^2\end{vmatrix}=k(abc)(a+b+c)^3$$ Find $k$.
If I directly open the determinant it will go to long I can't apply most of the row or column operation as they keep making it more complex.
| Let $$a=b=c=1$$ and you get the matrix
$$\begin{vmatrix}4&1&1\\1 &4&1 \\1&1& 4\end{vmatrix}=27k$$
The determinant is easily evaluated to be $54$ so $$27k=54$$.
Thus $$k=2$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3297816",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Where does $\pi^2$ appear spontaneously within Physical Phenomenon and Mathematics Equations? The term $\pi$ is found to appear in many equations and natural phenomenon; however my question is related to $\pi^2$.
While trying to figure out the reason for some $\pi^2$ terms appearing in certain equalities that I came a... | List of Places where π^2 can be seen-
*
*π is present in some structural engineering formulae, such as the buckling formula derived by Euler, which gives the maximum axial load F that a long, slender column of length L, modulus of elasticity E, and area moment of inertia I can carry without buckling
*The fact that ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3298039",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "24",
"answer_count": 14,
"answer_id": 0
} |
The integral $\int\limits_0^\infty\frac{x^4e^x}{(e^x-1)^2} \mathrm{d}x$ How to calculate the following integral $$\int_0^\infty\frac{x^4e^x}{(e^x-1)^2}\mathrm{d}x$$
I would like to solve this integral by means of two different ways: for example, integration by parts and using Residue Theorem.
| Using
$$ \frac1{(1-x)^2}=\sum_{n=0}^\infty(n+1)x^n $$
\begin{eqnarray}
\int_0^\infty\frac{x^4e^x}{(e^x-1)^2}\mathrm{d}x&=&\int_0^\infty\frac{x^4e^{-x}}{(1-e^{-x})^2}\mathrm{d}x\\
&=&\int_0^\infty x^4e^{-x}\sum_{n=0}^\infty(n+1)e^{-nx}\mathrm{d}x\\
&=&\sum_{n=0}^\infty\int_0^\infty(n+1)x^4e^{-(n+1)x}\mathrm{d}x\\
&=&\su... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3298116",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
What $x$ makes $\frac{x}{(a^2 + x^2)}$ maximum? The problem (Calculus Made Easy, Exercises IX, problem 2 (page 130)) is:
What value of $x$ will make $y$ a maximum in the equation
$$y = \frac{x}{(a^2 + x^2)}$$
I successfully differentiate, equate to zero, and wind up with
$$x^2 = a^2$$
Which gives me the answer of
... | We are given
$$y = \frac{x}{a^2 + x^2}$$
where $a$ is a constant.
Differentiating with respect to $x$ using the Quotient Rule yields
\begin{align*}
y' & = \frac{1(a^2 + x^2) - x(2x)}{(a^2 + x^2)^2}\\
& = \frac{a^2 + x^2 - 2x^2}{(a^2 + x^2)^2}\\
& = \frac{a^2 - x^2}{(a^2 + x^2)^2}
\end{align*}
Setting the deriva... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3298244",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 3
} |
Finding the inflexion points of a cubic in $ \mathbb{P}^{2}_{\mathbb{C}}.$
For what values of $ m $ is the cubic $$ F = x_{0}^{3} + x_{1}^{2} + x_{2}^{3} + mx_{0}x_{1}x_{2} = 0 $$ in $ \mathbb{P}^{2}_{\mathbb{C}} $ nonsingular? Find its inflexion points.
I know that $$ \text{Sing}(F) = \Big\lbrace F = \frac{\partial ... | You need to solve simultaneously the original equation times the Hessian.
Adding $6m^2$ times the original equation to the Hessian gives
$$(216+8m^3)x_0x_1x_2=0.$$
Unless $m^3=-27$ then $x_0x_1x_2=0$ so one of the variables vanishes.
If $x_0=0$ then $x_1^3+x_2^2=0$ so you get three inflection points
$(0:1:-\zeta)$ wher... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3298331",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Sliding Motion and Fillippov system I have problems to understand how works Fillipov system (see page 2 and 3) : So to make things easier, let consider the example $$\dot x=-\text{sgn}(x),\tag{E}$$ where $\text{sgn}(x)$ is the sign function (i.e. is 1 if $x>0$ and $-1$ if $x<0$).
So indeed the vector field $f(x)=-\text... | In the example equation, considered as a conventional ODE, the domain is the largest open set so that the right side is continuous, which is the real line without zero. A conventional solution is $x(t)=x_0-t$ if $x_0>0$. But it only exists for $t<x_0$ before leaving the domain of the ODE. There also does not exist a su... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3298459",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Is it legal to say $f(E)=\emptyset$ if set $E$ not in the function $f$ domain In many inverse functions, I have seen $f^{-1}(E)=\emptyset$, where the set $E$ is not in the function $f$ range.
So, is it also right to say $f(E)=\emptyset$ if set $E$ not in the function domain.
| The inverse image notation is very much standard; in most contexts, it may be used without clarification. The image of a set under a function is almost as standard. Given a function $f : X \to Y$, when seeing $f(E)$, it is typically understood that $E$ is a subset of $X$.
That said, the convention of saying $f(E) = \em... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3298688",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 1
} |
Does an integral of the form $\int f(x) \, \sqrt{dx}$ have any meaning? If $f(x)$ is a Riemann-integrable function, what meaning is there to an integral of the form $$\displaystyle\int f(x) \, \sqrt{dx}~?$$ I have read that stochastic processes like Brownian motion may be described by integrals of somewhat unusual form... | Good question! Think of differentials like $dx^2$ existing because integrals undo derivatives:
$$\begin{align}
\frac{d}{dx}\frac{d}{dx}\,f(x) &= f’’(x) \\[2ex]
\frac{d^2f}{dx^2} &= f’’(x) \\[2ex]
d^2f &= f’’(x)\, dx^2 \\[2ex]
\end{align}$$
Let’s integrate once (ignoring the $+c$). Remember how integrating always elimin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3298827",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
If $x,y \in V$ are linearely independent, then there exists a transvection $\tau$ with $\tau(x)=y$ Let $V$ be a $n$-dimensional $K$ vector space. A $\tau \in \operatorname{GL}(V)$ is called a transvection, if there exists a $(n-1)$-dimensional $\tau$-invariant subvector space $W$ of $V$ with
$$\tau_{W} = id_W \text{ a... | Since $\{ x, y \}$ are linearly independent, we also have that $\{ x - y, x \}$ are linearly independent. Complete this set to a basis
$$ w_1 = x - y, w_2, \dots, w_{n-1}, w_n = x $$
for $V$ and set $W = \operatorname{span} \{ w_1, \dots, w_{n-1} \}$. Define a linear map $\tau \colon V \rightarrow V$ by requiring that
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3299068",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Evaluate $\int_{0}^{\infty} \frac{\sin x-x\cos x}{x^2+\sin^2x } dx$ The integral $$\int_{0}^{\infty} \frac{\sin x-x\cos x}{x^2+\sin^2x } dx$$ admits
a nice closed form. The question is: How to evaluate it by hand.
| $$ I=\int_{0}^{\infty} \frac{\sin x-x \cos x}{x^2+\sin^2 x} dx= - \int_{0}^{\infty}\frac{\frac {x\cos x -\sin x}{x^2}}{1+(\frac{\sin x}{x})^2}dx=
-\int_{1}^{0} \frac{dt}{1+t^2}=\frac{\pi}{4}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3299202",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Can we deduce two contradictory statements from a wrong statement? In the proof that $\sqrt{p}$ is irrational where $p$ is a prime number:
We first assume $\sqrt{p}$ is rational.
From this we deduce $\sqrt{p}=\dfrac{a}{b}$, where $a$ and $b$ are co-prime.
Then using other reasonings we deduce $a$ and $b$ are not co-pri... | The statement that $\sqrt{p}$ (with $p$ a prime number) is rational is not just a false statement, but a statement that contradicts the basic axioms for the real numbers. Indeed, it is from the statement that $\sqrt{p}$ is rational together with those axioms that you derive an explicit contradiction.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3299406",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 4
} |
Proving $\frac{\sqrt{k(k+1)} }{k-1} \leq1 + \frac{2}{k} + \frac{3}{4k^2}$ for $k \geq 3$. Could you please give me a hint on how to prove the inequality below for $k \geq 3$?
$$\frac{\sqrt{k(k+1)} }{k-1} \leq1 + \frac{2}{k} + \frac{3}{4k^2} $$
Thank you in advance.
| You have to solve this inequality system:
$$\left\{\begin{matrix}
& \\k \neq 1 \: \land \: k\neq0
& \\k(k+1)\geqslant0
& \\1+\frac{2}{k}+\frac{3}{4k^2}\geq 0
& \\ \frac{\sqrt{k(k+1)}}{k-1} \geq 0
& \\\left (\frac{\sqrt{k(k+1)}}{k-1}\right )^2\leq \left ( 1+\frac{2}{k}+\frac{3}{4k^2}
\right )^2
\end{matrix}\righ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3299532",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Conjugate transpose of matrix is the adjoint intuition I'm having a bit of trouble understanding this fact from Linear Algebra Done Right
Let T $\in \mathcal{L}(V, W)$ Suppose $e_{1}, \dots, e_{n}$ is an
orthonormal basis of $V$ and $f_{1}, \ldots, f_{m}$ is an orthonormal
basis of $W$. Then then adjoint $T^{*}$ i... | Any inner product $\langle v, w\rangle$ on a real vector space has a representation as a symmetric positive-definite matrix $M$, with
$$\langle v, w\rangle = v^T M w.$$
Now if $T$ and $T^*$ are adjoint you must have
$$\langle Tv, w\rangle = v^T T^TMw = v^T MT^* w = \langle v, T^*w\rangle.$$
Since $v$ and $w$ are arbitr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3299625",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
The rank of a symmetric matrix equals the number of nonzero eigenvalues. I am wondering why
the rank of a symmetric matrix equals its number of nonzero
eigenvalues.
I have tried showing it like this:
A symmetrix matrix A can be written:
$$A=PDP^T$$, where P is an orthogonal matrix.
It is not difficult to see that ... | More precisely, I would say that the rank of a symmetric matrix is equal to the sum of the geometric multiplicities of its nonzero eigenvalues.
For example, If $A$ has two nonzero eigenvalues, say $2$ and $3$, with geometric multiplicities $1$ and $2$ respectively, then the rank of $A$ is $1+2 = 3$.
In this example, $2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3299724",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
1025th term of the sequence $ 1,2,2,4,4,4,4,8,8,8,8,8,8,8,8, ... $ Consider the following sequence - $$ 1,2,2,4,4,4,4,8,8,8,8,8,8,8,8, ... $$
In this sequence, what will be the $ 1025^{th}\, term $
So, when we write down the sequence and then write the value of $ n $ (Here, $n$ stands for the number of the below term) ... | Make up the frequency and cumulative frequency table:
$$\begin{array}{c|c|c}
x&f&F\\
\hline
1&1&1=2^1-1\\
2&2&3=2^2-1\\
4&4&7=2^3-1\\
8&8&15=2^4-1\\
\vdots&\vdots&\vdots\\
256&256&511=2^{9}-1\\
512&512&1023=2^{10}-1\\
1024&1024&2047=2^{11}-1\\
\vdots&\vdots&\vdots\\
2^n&2^n&2^{n+1}-1
\end{array}$$
So, your approach was... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3299825",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 7,
"answer_id": 6
} |
Jordan form of operator $X \mapsto AXA$ Matrices $n \times n$ on complex field. Compute Jordan form of operator $X \mapsto AXA$:
$$
A = \begin{bmatrix}
0 & 1 & & \\
& 0 & \ddots & \\
& & \ddots & 1 \\
& & & 0
\end{bmatrix}
$$
A is nilpotent Jordan block
| Hint: The Jordan form of the map $T(X) = AXA$ can be deduced using (only) the following pieces of information:
*
*$T$ is a linear map on a space with dimension $n^2$
*$T^n = 0$
*More generally, $\operatorname{rank}(T^{k}) = (n-k)^2$, $k = 1,\dots,n$
Another approach: using the vectorization operator, we can conclu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3299984",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Trying to Understand the Meaning of Transitivity in Relation to a Particular Problem I was trying to understand transitive relation and so I was solving a problem. The question is : $R_1 = \{(a,b)| a =b \text{ or }a = -b\} , R_2 = \{(a,b)| a =b \}, R_3 = \{(a,b)| a =b+1\}$, which one is transitive and why?
As far as I ... | Try replacing, for example, your $a \gt b$ with $\{a,b\} \in R_1$, $b \gt c$ with $\{b,c\} \in R_1$ and $a \gt c$ with $\{a,c\} \in R_1$. Then replace $R_1$ with $R_2$ and $R_3$ to see for which of these the first $2$ statements (e.g., if $\{a,b\} \in R_1$ and $\{b,c\} \in R_1$) means the third one must hold as well (e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3300104",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
If $B(x, r)$ is closed in $S\subseteq \mathbb{R}$, is it closed in $\mathbb{R}$? Let $(\mathbb{R}, d)$ a metric space. Let $S \subseteq \mathbb{R}$.
I know that an open ball in $S$ is not necessarily an open ball in $\mathbb{R}$, but is it a closed ball in $S$ closed in $\mathbb{R}$?
| No. Consider $S = (-1,1)$. Then $\overline{B_S}(0,1) = S$ which is not closed in $\mathbb{R}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3300229",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
invert one column before matrix multiplication and multiply result with unit vector = still same ranking, why? I am currently trying to find a mathematical proof of the following for a research paper. It's been quite long since i did higher mathematics and english is not my first language, so go easy on me if i got the... | I found a more or less formal proof for the observed phenomenon through splitting up the individual elements in the matrix, similar to this:
k is zxh; i is hxe; u is zxe; t is 1xe
Before:
$$ u_{11}^v = k_{11}*i_{11} + k_{12}*i_{21} + k_{13}*i_{31} + ... $$
After:
$$ u_{11}^n = -k_{11}*(1-i_{11}) + -k_{12}*(1-i_{21}) + ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3300332",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Finding the number of solutions to $\cos^4(2x)+2\sin^2(2x)=17(1+\sin 2x)^4$ for $x\in(0,2\pi)$
Number of solution of the equation
$\cos^4(2x)+2\sin^2(2x)=17(1+\sin 2x)^4\; \forall $ $x\in(0,2\pi)$
what i try
$\cos^4(2x)+2\sin^2 2x=17(1+\sin^2(2x)+2\sin 2x)^2$
$1+\sin^4 (2x)=17(1+\sin^4 2x+2\sin^2 2x+4\sin^24x+4\sin 2... | You're not required to find all solutions, just to find how many there are.
Let $u=\sin(2x)$. Then the trigonometric equation in $x$ becomes a polynomial equation in $u$:
$$
0 = (1 - u)^4 + 2 u^2 - 17 (1 + u)^4 = -2 (8 u^4 + 36 u^3 + 47 u^2 + 36 u + 8)
$$
Now plot this function of $u$ and see how many solutions are in ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3300465",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Integrating $\int (x+1)^2 dx$ two ways gives different results: $\frac13 x^3+x^2+x$ vs $\frac13 x^3+x^2+x+\frac13$. Why? I was trying to compute $$\int (x+1)^2 \, dx~,$$ which is a really easy function to integrate.
But the thing is that I write the function as $x^2 +2x+1$ and the result I got was $\frac{x^3}{3} +x^2 ... | WARNING while dealing with INDEFINITE INTEGRALS!!
When you solve some indefinite indefinite integral you should always add a constant in your final result (why?). Remember that indefinite integral is nothing but anti-derivative. Therefore, whenever you integrate you miss a constant which disappears because the derivati... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3300660",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 1
} |
Prove : $C_{k}(x)+C_{k+1}(x)\geqslant 1$ , $x_{k}\leqslant x\leqslant x_{k+1}$ Suppose $-\infty < x_{1}< x_{2}< \cdots < x_{n}< +\infty (n\geqslant 2)$ . And suppose a algebraic polynomial $C_{k}(x)$ $(k=1,2,\cdots ,n)$ ($degree\leqslant n-1$ ) satisfy :
$$C_{k}(x_{i})=\left\{\begin{matrix} 0, & i\neq k,\\ 1, & i=k, \... | As the answer by Dunham has stated, the solution involves several aspects. First, let
$$P_k(x) = C_k(x) + C_{k+1}(x), \text{ for } 1\leqslant k\leqslant n-1 \tag{1}\label{eq1}$$
As you've indicated, you get
$$C_j(x) = \frac{\prod_{i=1,i\neq j}^{n}(x - x_i)}{\prod_{i=1,i\neq j}^{n}(x_j - x_i)}, \text{ for } 1 \le j \le ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3300763",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Find a point that minimizes the sum squared difference of the distances to a set of other points I have n points in Euclidean space $\{\mathbf{a_1}, \mathbf{a_2}, ... , \mathbf{a_n}\}$, and the desired distances to them $\{d_1, d_2, ..., d_n\}$. How can I find the optimal point $\mathbf{x}$ that minimizes $\sum_{i=1}^n... | Using automaticallyGenerated's answer as a basis for the discussion, you need to minimize
$$\Phi(x,y)=\sum_{i=1}^{n} \left(\sqrt{(x-x_i)^2+(y-y_i)^2}-d_i\right)^2$$ which is highly nonlinear and "good" starting values are required.
You can get those easily if, in a prelimary step, you consider that you have $n$ equatio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3300898",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Which primes $p$ satisfy $n^2 \equiv -1 \mod p$ for a perfect square $n^2$? I am trying to solve a homework exercise in elementary number theory:
Which primes $p$ satisfy $n^2 \equiv -1 \mod p$ for a perfect square $n^2$?
After looking at the case $p=5$, I saw that $3^2 \equiv 4 \mod 5$, but $p=2^2+1$, I thought tha... | The answer is exactly the primes which are congruent to $1,2 \bmod 4$.
The only prime congruent to $2 \pmod 4$ is $2$, so assume $p$ is odd for the rest of the answer.
The result follows pretty straightforwardly once we establish a basic result about the ring $\mathbb{Z}/p\mathbb{Z}$.
Result: There exists some $g$ such... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3301017",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Monotone increasing bijection from $\mathbb{R}$ to $(0,1)$. Give an example of a monotone increasing function $f: \mathbb{R} \to (0,1)$ such that $f$ is a bijection.
I have an example in mind that follows $$g(x)=\frac{1}{1+e^x} ,x \in \mathbb{R}.$$ Then $g$ is a monotone decreasing bijection from $\mathbb{R}$ to $(0,1)... | Your example is correct.
You may take other bijective function as $$f:(-\infty,\infty) \rightarrow (0,1),
f(x)=\frac{1}{2}(1+\mbox{erf}(x)), \mbox{erf}(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x} e^{-t^2} dt,$$ then $$f'(x)=\frac{2}{\sqrt{\pi}} e^{-x^2}>0, f(\pm \infty)=\pm 1.$$
Another example is $$f:(-\infty,\infty) \rightar... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3301197",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Can anyone help showing why my calculation of $\int x\ln x$ $dx$ is wrong?
Suppose $\frac{dy}{dx}=x\ln x,$ my teacher asks me to find $y$.
So I assume I got to integrate the right hand side:
$$\int x\ln x\, dx$$
The result I got is
$$
\int x\ln x\, dx=x\ln x-x+C\tag{1}
$$
But, apparently, it is wrong since taking t... | Using Latex is tiring, so i will just use drawing.
Differentiating the answer will definitely get you into the function that you want to integrate earlier, im using Integration by parts with the table method
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3301319",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 1
} |
For $a>0$, prove $\int_{-\infty}^\infty e^{-x^2}\cos ax\,dx=\sqrt{\pi}e^{-\frac{a^2}{4}}$ by justifying term by term integration.
For $a>0$, prove $\displaystyle\int_{-\infty}^\infty e^{-x^2}\cos ax\,dx=\sqrt{\pi}e^{-\frac{a^2}{4}}$ by justifying term by term integration.
$$\int_{-\infty}^\infty e^{-x^2}\cos ax\,dx=\... | Since for any $n\in \mathbb N$,$$\left|e^{-x^2}\sum_{k=0}^{n}(-1)^k\frac{(ax)^{2k}}{(2k)!}\right|\leq e^{-x^2}\sum_{k=0}^{+\infty}\frac{(a|x|)^{2k}}{(2k)!}\leq e^{-x^2}\sum_{k=0}^{+\infty}\frac{(a|x|)^{k}}{k!}=e^{-x^2+a|x|},$$
and $$\int_{\mathbb R}e^{-x^2+a|x|}\,dx=2\int_0^\infty e^{-x^2+ax}\,dx<\infty,$$
we can use D... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3301437",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Integration of $x^2\cdot\frac{x\sec^2x+\tan x}{(x\tan x+1)^2}$ Integrate
$$\int x^2\cdot\dfrac{x\sec^2x+\tan x}{(x\tan x+1)^2}dx$$
So what is did is integration by parts taking $x^2$ as $u$ and the other part as $v$ . Now I got to use it again which then eventually leads to (integral of $\dfrac1{x\tan x+1}dx $). Can s... | $x \tanx + 1 = t$
$$\int x^2\cdot\dfrac{x\sec^2x+\tan x}{(x\tan x+1)^2}dx$$
$$=\int \frac{x^2}{t^2}dt = x^2* (-1/t) + \int (2x/t)dt$$ (By parts)
$$ = x^2(-1/t) + 2x \ln|t| -2(t \ln|t|-t) + C$$ (By parts again)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3301557",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Splitting Field of $x^4 + x^3 + 1$ over $\mathbb{F}_{32}$ I'm trying to find the splitting field described in the title. I believe I have figured it out, but my method seems a bit involved and I'm wondering if there is any simpler way to obtain the result.
My method:
This polynomial is actually contained in $\mathbb{F}... | Note that the lattice of fields of the shape $$\Bbb F_{\displaystyle 2^r}$$ corresponds to the lattice of the $r$-values w.r.t. division. The field $$\Bbb F_{32}=\Bbb F_{2^5}$$ intersects (in a common embedding) the fields $\Bbb F_{2^k}$ for $k=1,2,3,4$ only in $\Bbb F_2$, in the prime field.
The polynomial
$$ f=X^4+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3301686",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Maximizing the Sum of Cubes I have ten variables, $x_1$ through $x_{10}$. $-1 \leq x_i \leq 1$, and $\sum x_i = 0$. What is the maximum of $\sum x_i^3$? I've tried to use Lagrange multipliers, but writing the restricted domain as a constraint seems, if not impossible, then very difficult.
| You can write it in a standard form
\begin{equation}\notag
\begin{split}
\min & -\sum_{i=1}^{10} x_i^3 \\
\mathrm{s.t.} \hspace{1ex}& x_i - 1 \leq 0 \\
& -1 -x_i \leq 0 \hspace{1ex} \forall i \in [1, 10]\\
& \sum_{i=1}^{10} x_i = 0 \\
\end{split}
\end{equation}
Then the Lagrange Dual will be
\begin{equation}
L(x,\lamb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3301790",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Operator norm of translated operator Suppose $P$ is a compact operator in $L^2([0,1])$. Assume it's contractive in the sense that $\|P\|_{\mathrm{op}}<1$. For some $f$ in $L^2([0,1])$, define the operator
\begin{equation}Tg(x):= f(x) +Pg(x).\end{equation}
Is it true that $T$ remains a contraction? From my initial work,... | Let's make a simple example. Let $P$ be the halving operator, $g$ be the constant $1$ function, and $f$ be the constant $2$ function. Then
$$ T g = f + P g = 2 + \frac{1}{2} 1 = \frac{5}{2} > ||g|| = 1 \text{.} $$
The intuition is that if $f$ is "big", $f$ controls the norm of $Tg$. So then if $g$ is "small" com... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3302020",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Boundary-value problem for linear hyperbolic system by Fourier series I am trying to solve the linear equations
$$\partial_t \rho +\partial_x \varphi =0, \qquad \partial_t \varphi+\partial_x \rho = \alpha \rho +\beta \varphi,$$
where $\alpha$, $\beta$ are constants. The functions $\rho$, $\varphi$ are defined on $[0,T]... | Applying $\partial_t$ to $\varphi_t + \rho_x = \alpha\rho + \beta\varphi$ and using $\rho_t = -\varphi_x$ leads to
$$
\varphi_{tt} - \varphi_{xx} = \beta\varphi_t- \alpha\varphi_x \, .
$$
The corresponding initial conditions are $\varphi(0,x) = 0$ and $\varphi_t(0,x) = 0$, and the boundary conditions are $\varphi_x(t,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3302106",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
The derivative of $ \tan x$ is $ \sec^2 x$. Why? I understand why the derivative of $\sin x$ is $\cos x$, and why the derivative of $\cos x$ is $-\sin x$ in a geometric way.
But I can not understand why the derivative of $\tan x$ is $\sec^2 x$.
Can someone please explain this in a visual, geometric way using the unit... | First, look at the graph of $\tan(x)$.
It has vertical asymptotes at integer multiples of $\dfrac{\pi}{2}$ and is undefined at $-\dfrac{\pi}{2}$ and $\dfrac{\pi}{2}$. Observe that from $-\dfrac{\pi}{2} \leq x \leq \dfrac{\pi}{2}$, the slope of $\tan(x)$ is always increasing. Notice that it increases faster from $-\dfr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3302218",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 1
} |
How to find $x$ from expression?
Solve for $x$ in the expression $$x(2x + 5) = 168.$$
I have already tried to move $x$ from one side to another with brackets and without them, like this:
$x = 168;\quad 2x + 5 = 168 \implies 2x = 163 \implies x = 81.5$
but eventually without success.
I know that the answer is $x = 8$... | if $f(x)g(x) = A$ that does not mean that $f(x)=A$ or $g(x) = A$. However, what you could do is expand as follows
$$2x^2 + 5x = 168$$
then
$$2x^2 + 5x - 168 = 0$$
then you have to realize that what you have is a quadratic equation, where normally you compute a quantity called the discriminant
$$\Delta = b^2 - 4ac$$
for... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3302325",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
For independent r.v. the $\liminf X_n$ is almost surely constant I need to show that for independent real-valued r.v. $(X_n)_{n\in \mathbb{N}}$ the $X_*:= \liminf_\limits{n\rightarrow \infty}X_n$ is almost surely constant with $X_*\in\mathbb{R}\cup\{-\infty,+\infty\}$.
A r.v. $X$ is constant iff $X(\omega)=c$ for $\ome... | The event $\{X_{*} \leq x\}$ belongs to $\sigma (X_k,X_{k+1},...)$ for each $k$. By 0-1 law $P(X_{*} \leq x) =0$ or $1$ for each $x$. The distribution function $F$ take only the values $0$ and $1$ and so it jumps from $0$ to $1$ at some point $c$. ($c=\sup \{t: F(t) =0\}$). It follows that $X_{*}=c$ with probability $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3302568",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Area of parallelogram = Area of square. Shear transform Below the parallelogram is obtained from square by stretching the top side while fixing the bottom.
Since area of parallelogram is base times height, both square and parallelogram have the same area.
This is true no matter how far I stretch the top side.
In belo... | Behold, $\phantom{proof without words}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3302853",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "38",
"answer_count": 9,
"answer_id": 6
} |
Distance between two circles in a sphere I have a sphere with radius $R$ and $O$ is the origin.
Inside sphere there are 3 circles. The small circle in black colour is fixed with it's position defined by and $\alpha$ angle and among other two circles one is great circle and another small circle can rotate by maintaining... | According to the comments on the question, I will start by assuming that $\alpha$ and $\beta$ are given.
Let's name some additional points on the sphere. Let $C$ be the point $(0,0,R)$ where the positive $z$ axis intersects the sphere.
The blue great circle intersects the $y,z$ plane in two points;
from those two poin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3303052",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Plugging inequalities into equations Let's say I have the inequality $3f \leq 2m$. If I wanted to plug this into the equation $n-m+f=2$, how would I do that? Where would I begin?
| I assume that you can replace the $=$ sign with a $≤$ (less than OR equal to) so thus $n-m+f≤2$ adding gets $n-3m+4f≤2$
If your not sure of this use integers to test out this theory!!
Ex:$3≤3$ and $6≤7$ so $9≤10$
Notice this only works when there is a $≤$ involved
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3303183",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Are Hilbert Scmidt integral operators on separable compact Hausdorff spaces in the Hilbert Schmidt class? Let $X$ be compact separable Hausdorff space with a positive Borel measure $\mu$. Assume $L^2(X)$ is separable.
Consider a function $K: X \times X \to \mathbb{C}$ with $K \in L^2(X \times X , \mu \otimes \mu)$. We ... | This is true and , in fact, the converse is also true. Every H-S operator on $L^{2}(\mu)$ is of this type for some $K$. Reference: Theorem VI.23, p. 210, Functional Analysis, Vol 1 by Reed and Simon.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3303302",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to find the overall visible area of overlapping 3 or more Octagons? I have 3 Octagons of same size. I know the coordinates of their centers and their side length. How can I calculate the overall area? (Shaded area in the image)
In the image above, the area comes out to be 1343.73 sq units, calculated via a CAD sof... | You have enough information to find the vertices of each octagon.
A search for overlapping polygons area finds many links to versions of the the polygon clipping problem. Something there might solve your problem.
https://www.google.com/search?client=ubuntu&channel=fs&q=overlapping+polygons+area&ie=utf-8&oe=utf-8
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3303443",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Compactness and metric space i know that if A and B are compact then there exists $(a,b)\in A\times B, d(a,b) = d(A,B)$ I want to find an example where this is not true if A is compact and B closed
I put $A=[1,2]$ and $B=]-\infty,0[ $ in $\mathbb{R}^*=]-\infty,0[\cup ]0,+\infty[$
is it correct ?
here B is closed but no... | Yes, this is correct. In fact, you could have taken $B=[-1,0[$. While it is not closed in $\mathbb{R}$, it is closed in your $\mathbb{R}^*$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3303558",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
$2$ out of $4$ points, each of distance at most $1$ apart, are at most $1/\sqrt2$ apart Given $4$ points in the plane such that any pair of them are a distance of at most $1$ apart, show that some pair of them must be of distance at most $1/\sqrt{2}$ apart.
I figured the solution might involve a pigeonhole argument, b... | By eyeballfrog's comment, if there's an arrangement of four points such that all pairwise distances lie in the interval $(1/\sqrt{2}, 1]$, then by scaling, we can guarantee that two are a distance $1$ apart. Let these points be $A$ and $B$. The diagram shows circles of radius $1/\sqrt{2}$ and $1$ with centers $A$ and $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3303675",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
antiderivatives of $\frac1{x-1}$ I was told to find the anti-derivative of this problem: $$\int\frac{2x^2-13x+18}{x-1} dx$$
I solved the problem - first dividing and then finding the anti-derivative: $$x^2-11x+7(\ln(x-1))+c$$
The given answer was the same as mine, but they put the $x-1$ in $\ln(x-1)$ in absolute value ... | Hint: Since we have $$\int\frac{1}{x}dx=ln(|x|)+C$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3303776",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Finding the maximum without differentiation
I have this problem that needs to be solved as if I was a GCSE student.
After cutting a square of length $x$ from each corner, the volume of the open box would be $V= x(10-2x)(10-2x)$. We want to find the value of $x$ so that this volume is maximised. The ordinary solutio... | Note that by AM-GM inequality,
$$4V=4x(10−2x)(10−2x)\leq \left(\frac{4x+(10−2x)+(10−2x)}{3}\right)^3=\left(\frac{20}{3}\right)^3.$$
Equality holds, and the maximum value of $V$ is attained, when $4x=10-2x$, i.e. for $x=5/3$.
P.S. Actually what you need is just a particular case of AM-GM inequality: for $a,b\geq 0$,
$$a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3303846",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to show that $\sqrt{2+\sqrt{3}} = \dfrac{\sqrt{6}+\sqrt{2}}{2}$
$\sqrt{2+\sqrt{3}} = \dfrac{\sqrt{6}+\sqrt{2}}{2}$
How to change $\sqrt{2+\sqrt{3}}$ into $\dfrac{\sqrt{6}+\sqrt{2}}{2}$
| Hint: $$\left( \frac{\sqrt{6} + \sqrt{2}}{2}\right)^2 = \frac{6 + 2 \sqrt{6} \sqrt{2} + 2}{4} = ?$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3303949",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 2
} |
Evaluating $\sqrt{a\pm bi\sqrt c}$ I recently encountered this problem
$$\sqrt{10-4i\sqrt{6}}$$
To witch I set the solution equal to $a+bi$ squaring both sides leaves
$${10-4i\sqrt{6}}=a^2-b^2+2abi$$
Obviously $a^2-b^2=10$ and $2abi=-4i\sqrt{6}$, using geuss and check, the solution is $a=\sqrt12, b=\sqrt2$
But I was wo... | We have that
$$a^2-b^2=10 \quad \textrm{and} \quad 2ab=-4\sqrt 6$$
Now, squaring both equalities and addem up we get
$$(a^2+b^2)^2=(a^2-b^2)^2 +(2ab)^2=10^2+(-4\sqrt 6)^2 =196$$
$$\Rightarrow \quad a^2+b^2=14$$
and using again the first equality we obtain
$$a^2=12 \quad \textrm{and} \quad b^2=2$$
or
$$a=\pm 2\sqrt 3 \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3304125",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
On the inclusion homomorphism $\mathbb Z\to\mathbb Q$ This question is about the group homomorphism $i: \mathbb Z\to\mathbb Q$ given by $m\mapsto m$.
1) Is the following proof of the fact that the arrow $i$ in $\mathbf {Ab}$ is epic correct? Let $h,h':\mathbb Q\to G$ be group homomorphisms. Suppose $h\circ i=h'\circ i... | The inclusion $\mathbb{Z} \hookrightarrow \mathbb{Q}$ is not epic in $\mathbf{Ab}$, but it is epic in $\mathbf{Ring}$ as well as in the category of torsion-free abelian groups.
Also, there is no nonzero homomorphism from $\mathbb{Q}$ to $\mathbb{Z}$, because the image of any rational number would have to be an integer ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3304227",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
Maximal ideal and not algebraically closed field This is a question concerning maximal ideals in a polynomial ring over a non-algebraically closed field k.
First is the example inspired for this question: as a standard exercise, it is easy to show that $\langle x^2+1\rangle$ is a maximal ideal in $\mathbb{R}[x]$ (Brief... | The Hilbert Nullstellensatz for non-algebraically closed fields describes
all maximal ideals $I$ of $R=k[x_1,\ldots,x_n]$. It states that $R/I$ is a finite
field extension of $k$. This means that $I$ is the kernel of a
$k$-algebra homomorphism $\phi:R\to k^{\text{alg}}$. Such a map is given by
$\phi(f(x_1,\ldots,x_n))=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3304318",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Does the logarithm satisfy any differential equation? The exponential function $\exp:\mathbb{R}\to\mathbb{R}_+$ satisfies the differential equation $f^\prime(x)=f(x)$.
Does the logarithm $\log:\mathbb{R}_+\to\mathbb{R}$ also satisfy any differential equation?
Curious about if this gets closed immediately or not... seem... | Here's a fun way to see the relationship between the definition for the exponential function and the "obvious" answer:
Suppose $y = e^x$. By definition of the exponential function, this means that $dy/dx = y$. Now view $x(y)$ as a function of $y$, i.e., $x$ is the inverse of the exponential function. Rearranging the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3304424",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 6,
"answer_id": 3
} |
Does $T$ bounded linear operator necessarily imply weak sequential continuity Let $X,Y$ be Banach Spaces, show that $x_{n} \xrightarrow{w} x$ and $T \in BL(X,Y)\Rightarrow Tx_{n} \xrightarrow{w} Tx$
Question: Does sequential continuity (which $T$ clearly has) necessarily imply that $T$ is weak-sequentially continuous? ... | If $y^{*} \in Y^{*}$ then $x^{*}(x)=y^{*}(Tx)$ defines a continuous linear functional on $X$. Hence $y^{*}(Tx_n)\to y^{*}(Tx)$. This implies that $Tx_n \to Tx$ weakly.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3304542",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Why the normalizer of the Sylow $p$-subgroups of the symmetric group of degree $p$ has order $p(p-1)$ and is known as Frobenius group $F_{p(p-1)}$?
Why the normalizer of the Sylow $p$-subgroups of the symmetric group of degree $p$ has order $p(p-1)$ and is known as Frobenius group $F_{p(p-1)}$?
I am trying to unders... | My approach is more combinatorial than algebraic. Any group isomorphic to $(\mathbb Z_p)^n$ can be written as a permutation group on $p^n$ points, with any Frobenius complement a group that fixes exactly one of these points, therefore, maximally, a group of order $p^n - 1$. It can be shown that this maximal group alway... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3304678",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Cross-ratio concept from Euclidean Geometry I have seen on the following Wikipedia website the definition of "cross-product." Apparently, if four lines passing through a common point $P$ are traversed by a line and the points of intersection "in one direction" are A, B, C, and D, their cross-product is
\begin{equation*... | The cross-ratio of four points $A,B,C$ and $D$ on a line $l$ in Euclidean space is defined as the ratio of the ratios $AC:BC$ and $AD:BD$. Simplifying the double quotient yields
$$ [A,B;C,D] = \frac{\frac{AC}{BC}}{\frac{AD}{BD}} = \frac{AC\cdot BD}{AD \cdot BC}.$$
The semicolon in Wikipedia's notation indicates the dif... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3304826",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Expected Profit from a Lottery Ticket
Ahmed is playing a lottery game where he must pick 2 numbers from 0 to
9 and then one letter out of the 26-letter English alphabet. He may
choose the same number both times. If his ticket matches the 2 numbers
and 1 letter drawn in order, he wins the grand prize and receives... | Yes, but you are missing something. Can you figure it out?
$$2\cdot \frac{1}{26}\frac{1}{9}\frac{1}{10} + \frac{1}{26}\frac{9}{10} \frac{9}{10} $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3304959",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Conditions required for Galois Correspondence This question is based on Theorem 10.2 of John Howie's 'Fields and Galois Theory', p.171. All fields have characteristic zero.
$M$ is a normal radical field extension of $K$, i.e. $M=K(\alpha_1,\alpha_2,...,\alpha_n)$ where $\alpha_i^{p_i}\in K(\alpha_1,...\alpha_{i-1})$ fo... | You assume that $M/K$ is a normal extension. This means there is a polynomial $f\in K[x]$ such that $M$ is a splitting field of $f$ over $K$. But then note that $P$ is a splitting field of $(x^{p_i}-1)f(x)\in K[x]$, and hence $P/K$ is normal.
Note that in general if $P/M$ and $M/K$ are both normal extensions it doesn'... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3305075",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Describe the equalizer of $f,1:X\to X$ as explicitly as possible Exercise 5.2.22 from Leinster asks to describe the equalizer of $f,1:X\to X$ in $\mathbf{Set}$, where $f:X\to X$ is a map, as explicitly as possible.
What level of explicitness is expected? By definition, it is a pair $(S,h)$ where $S$ is a set and $S\to ... | The equalizer of $s,t:X\to Y$ in $\mathbf{Set}$ is $(E,i)$ where $E=\{x\in X:s(x)=t(x)\}$ and $i:E\to X$ is the inclusion map. This is clearly a fork, and if $(A,g)$ is another fork, then define $\bar g:A\to E, a\mapsto g(a)$. Note that $g(a)$ indeed lives in $E$ because $sg=tg$, and $\bar g$ is the unique map $A\to E$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3305168",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
How to calculate $\lim_{x \to - 1} \frac{2}{(x+1)^4}$ Calculate $$\lim_{x \to - 1} \frac{2}{(x+1)^4}$$
a) $0$
b) $\infty$
c) $-\infty$
d) $2$
I am able to see that it is equivalent the limit as $x$ approaches $-1$ of $\frac{2}{(x^2+2x+1)^2}$.
I know that when doing limits to infinity this would be $0$ because the deno... | Consider this:
$$ x\to-1\qquad\Leftrightarrow\qquad (x+1)\to 0$$
$$ \lim_{x\to-1}\frac{2}{(x+1)^4} = \lim_{(x+1)\to 0} \frac{2}{(x+1)^4} = \lim_{y\to 0} \frac{2}{y^4}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3305293",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 2
} |
what is the value of $\sum_{ijk}\omega u_iu_j \delta_{ik}\delta_{kj}$? I have to work with the following sum:
$\sum_{ijk}\omega u_iu_j \delta_{ik}\delta_{kj}$ where $\omega$ is a constant in $\mathbb{C}$.
Is the answer:
$$\omega \sum_k \sum_i u_i \delta_{ik}\sum_j u_j\delta_{kj}=\omega\sum_k u_k^2$$
or:
$$\sum_{ij}u_iu... | When you manipulate this expression:
$$\sum_{i, j} u_i u_j \left( \sum_k \omega \delta_{ik} \delta_{kj} \right),$$
you first (rightfully) consider the sum over $k$:
$$\sum_k \omega \delta_{ik} \delta_{kj}.$$
A product of real numbers is non-zero if and only if each of the terms is non-zero. Here, if $\delta_{ik}=0$ or ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3305497",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to integrate $\int_0^{2\pi} \cos^{10}\theta \mathrm{d}\theta$ using complex analysis. I am asked to evaluate the following integral:
$$\int_0^{2\pi} \cos^{10}\theta \mathrm{d}\theta$$
I am using complex analysis. Setting $z = e^{i\theta}$, I get from Eulers formula:
$$\cos \theta = \frac{1}{2}\left(e^{i\theta} + e^... | Hint
Set $$f(z)=\frac{(z+z^{-1})^{10}}{2^{10}iz}=\frac{(z^2+1)^{10}}{2^{10}iz^{11}}.$$
Then, $$f(z)=\frac{1}{iz^{11}}\sum_{k=0}^{10}\binom{10}{k}z^{2k}=...+\frac{1}{i2^{10}z}\binom{10}{5}+...$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3305633",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Number of ways to chose 6 courses out of 15 (different) courses We got 3 English courses, 6 Chinese courses, 6 Spanish courses.
Each course is different.
In how many was can we choose 6 courses such that we must chose atleast 1 course from each topic (English, Chinese, Spanish)
If possible, I would like to see a soluti... | There are some mistakes in your calculation. Fist, there is no reason to divide by $3!$. When you choose an English course, a Chinese course, and a Spanish course, the order matters. You haven't counted anything $6$ times. Second you have some double counting. Choosing English 1 and one of the first three courses... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3305708",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
A sum of products of binomial coefficients I am looking for a proof or a reference for the following
( apparent ) combinatorial identity:
$$
\sum_{i = s}^{s+t}\left(\,{-1}\,\right)^{\, i}{i \choose s}
{s \choose i - t}
=
\left(\,{-1}\,\right)^{s + t},\quad\mbox{where}\ s\geq t\geq 0\
\mbox{are integers}
$$
Any help wil... | The hint of @darijgrinberg is valuable and deserves an answer by its own.
We obtain
\begin{align*}
\color{blue}{\sum_{q=s}^{s+t}}&\color{blue}{(-1)^q\binom{q}{s}\binom{s}{q-t}}\\
&=\sum_{q=0}^t(-1)^{q+s}\binom{q+s}{q}\binom{s}{q+s-t}\tag{1}\\
&=(-1)^s\sum_{q=0}^t\binom{-s-1}{q}\binom{s}{t-q}\tag{2}\\
&=(-1)^s\binom... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3305804",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
$\zeta (s+c)/\zeta (s)$ is increasing It appears that for the Riemann zeta function
$$\zeta (s) = \sum_{n\ge 1} \frac{1}{n^s} \quad (s > 1)$$
and any constant $c > 0$, the function
$$\frac{\zeta (s+c)}{\zeta (s)}$$
(considered for real $s > 1$) is increasing.
Could anybody give a proof of this fact?
Thank you very much... | Since $s>1$, $\zeta(s)>0$ and it's easier to consider the logarithmic derivative: $f'(x)/f(x)$ has the same sign as $f'(x)$. In particular, for $s>1$ we can use the Euler product
$$ \zeta(s) = \prod_{p} (1-p^{-s})^{-1} : $$
from this we have
$$ \frac{d}{ds} \log{\zeta(s)} = \frac{d}{ds} \sum_p -\log{(1-p^{-s})} = -\su... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3306016",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Solving for the height of water in a sphere of radius $r$ being filled at a constant rate. I'm attempting to find a function that will output a value for the height $h$ (which can also be represented as $y$) of water in a sphere of radius $r$ being filled at a constant rate.
For finding the volume of a sphere until a c... | Cardano is not the best way to go here. Try the trigonometric solution. Your equation is already depressed, so you can ignore that part. Your equation has $$p = -3r^2\\q = \frac{3V - 2r^3\pi}{3\pi}$$$p < 0$, so the solutions will involve real numbers only.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3306120",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Finding a matrix with a given rowspace My linear algebra textbook asks,
Find a matrix $A$ that has $V$ as its row space if $V$ is the subspace spanned by
$$\begin{align}\begin{bmatrix}1\\1\\0\end{bmatrix},\begin{bmatrix}1\\2\\0\end{bmatrix} \text{, and}\begin{bmatrix}1\\5\\0\end{bmatrix}\end{align}$$
(Strang 4e, pr... | You are right. In fact $(1,1,0)^t$ and $(1,2,0)^t$ form a linearly independent set, and so, you can also put the matrix as
$$A = \begin{pmatrix}
1 & 1 & 0 \\
1 & 2 & 0
\end{pmatrix}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3306238",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Minimize variance of $aX + bY + cZ$ for independent variables $X,Y,Z$ There are three independent random variables $X, Y, Z$.
The goal is to minimize the variance of $aX + bY + cZ$ where $a+b+c = 1$ and $0\le a,b,c \le1$.
$$\operatorname{var}(X) = 2\operatorname{var}(Y) = 3\operatorname{var}(Z)$$
I did $$K = aX + bY + ... | To minimize $6a^{2}+3b^{2}+2c^{2}$ subject to $a,b,c \geq 0$ and $a+b+c=1$ note that $1^{2}=(a+b+c)^{2}=(\frac 1 {\sqrt 6}\sqrt 6 a +\frac 1 {\sqrt 3}\sqrt 3 a +\frac 1 {\sqrt 2}\sqrt 2 a)^{2} \leq (\frac 1 6 +\frac 1 3+\frac 1 2) (6a^{2}+3b^{2}+2c^{2})$ by C-S inequality. Hence $6a^{2}+3b^{2}+2c^{2} \geq 1$ and equal... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3306349",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Finding $P(X<2Y)$ given joint pdf $f(x, y) = \frac {1}{2\pi} e^{-\sqrt{x^2 + y^2}}$ for $x,y\in\mathbb R$
The joint pdf of $X$ and $Y$ is $$f(x, y) = \frac {1}{2\pi} e^{-\sqrt{x^2 + y^2}}\,; \quad x,y\in \mathbb{R}$$ Find $P(X<2Y)$.
I have tried this:
$$\int_{-\infty}^{\infty}\int_{-\infty}^{2y} f(x, y)\, dx\,dy$$
@S... | The joint distribution of $(X,Y)$ is rotationally symmetric, since the density depends only on the distance from the origin. This means that the conditional distribution of $(X,Y)$ given $X^2+Y^2$ is uniformly distributed on the circle of radius $r=\sqrt{X^2+Y^2}$. Writing $X=r\cos\Theta$ and $Y=r\sin\Theta$ with $\The... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3306478",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
When Is the Squeeze(Sandwich) Theorem Used? I am beginning to learn Calculus 1, and I was taught about the squeeze (sandwich) theorem. It seems to me that all the problems given have $\sin$ involved. Is this true? When is the squeeze theorem applied?
Also, what are the standard "squeeze functions"? For example, I kno... | No it's not necessary that sine functions are involved in all Sandwich Theorem Problems.
*
*Sandwich Theorem is commonly used in computing Integrals as a limit of a sum.
*It is used in Limit Computations
*It is used in proving convergence of many series by bounding it.
*I found another interesting use with many a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3306633",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Infinite Series $\sum_{n=1}^{\infty}\frac{4^nH_n}{n^2{2n\choose n}}$ I am trying to find a closed form for this infinite series:
$$ S=\sum_{n=1}^{\infty}\frac{4^nH_n}{n^2{2n\choose n}}$$
Whith $H_n=\sum\limits_{k=1}^{n}\frac{1}{k}$ the harmonic numbers.
I found this integral representation of S:
$$S=2\int_{0}^{1}\frac{... | From here, we have
$$\frac{\arcsin z}{\sqrt{1-z^2}}=\sum_{n=1}^\infty\frac{(2z)^{2n-1}}{n{2n \choose n}}$$
substitute $z=\sqrt{y}$, we get
$$\sum_{n=1}^\infty\frac{4^ny^n}{n{2n \choose n}}=2\sqrt{y}\frac{\arcsin\sqrt{y}}{\sqrt{1-y}}$$
Now multiply both sides by $-\frac{\ln(1-y)}{y}$ then integrate from $y=0$ to $1$ an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3306742",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
} |
How to reason that $n^5 - n$ is divisible by 2 as proof for a consequence of Fermat's little theorem. In my text book on Discrete Mathematics (I), we have a chapter that covers a bit of elementary Number Theory. In it we see the famous Theorem of Euler as well as the derived little theorem of Fermat. I understand these... | This is trivial:
*
*If $n$ is odd, is $n^5-n$ odd or even?
*If $n$ is even, is $n^5-n$ odd or even?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3306833",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 7,
"answer_id": 0
} |
The first 3 terms of the expansion of $\left(1+\frac{x}{2}\right)\left(2-3x\right)^6$ According to the ascending powers of $x$, find the first $3$ terms if the expansion of
$$\left(1+\frac{x}{2}\right)\left(2-3x\right)^6$$
For the expansion of $$(2-3x)^6 $$
The first 3 terms are $$64 -576x + 2160x^2$$
Now, are the req... | Add the two together, and keep the terms with $x^0$, $x^1$, and $x^2$. So the answer is $$64+(32-576)x+(2160-288)x^2$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3307055",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Finding the Maximum of a Continuous Function over a Closed Interval For function $f\left ( x \right )=4x^{3}-6x^{2},$ the maximum occurs in the interval $\left [ 1,2 \right ]$ when $x$ is equal to ___________
I got $x=0$ is maxima. Because, in that point $f''(x)<0$
But in answer, it is given though $f''(x)<0$ at $x=0,... | You are told to find the maximum value of $f(x) = 4x^3 - 6x^2$ in the interval $[1,2]$. So why did you simply assert $x = 0$ when this value is clearly not in the requested interval?
This is a common mistake students make: the calculation goes flawlessly, but there is no understanding of what it means.
When searching... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3307138",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Is a parallel translation a linear transformation? I guess not because a linear transformation maps a zero vector to the zero vector but parallel translation does not. Am I right?
| You are entirely correct. For any linear transformation $L$, it is true that $L(0)=0$, but this is not true for translation. This is enough to prove that translation is not a linear transformation.
If you want to go into detail, you can also go down to the definitions and find the axiom on which translation fails. Reme... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3307196",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Method of Characteristics - Lagrange-Charpit Equations I need to solve the following PDE with initial condition $U(x,0)=U_0(x)$. Once this is one of my first times, I'd like to get second opinions.
Many Thanks.
\begin{equation} \partial_t U + (\text{cos}(t)+1)\partial_x U = -2,
\end{equation}
The characteristic curve... | Note that while $U(x,0) = U_0(x)$, the PDE itself isn't satisfied.
\begin{align}
U_t + (1 + \cos t)U_x &= -1 + U_0'(x - t + \sin t)(-1 + \cos t) + U_0'(x - t + \sin t)(1 + \cos t) \\
&= -1 + 2\cos t \ U_0'(x - t + \sin t) \\
&\not \equiv -2.
\end{align}
But
$$
\frac{\mathrm{d}U}{\mathrm{d}t} = -2 \implies U(x(t),t) = -... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3307616",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
What's my mistake in finding $\int_0^\infty dx e^{-ax^2} \sin(b/x^2)$? I want to evaluate $$I=\int_0^\infty dx e^{-ax^2} \sin(b/x^2)$$ for $a,b>0$.
A first simplification is to substitute $y=x/\sqrt{a}$ and define $c=ab>0$ to obtain
$$I=\frac{1}{\sqrt{a}} \int_0^\infty e^{-x^2} \sin(c/x^2)$$
Now my idea was to use the ... | The Glasser's master theorem is a useful tool for the solution. First, use Euler's formula to decompose the sine term into the sum of exponentials. Then it boils down to computing the integral of the form
$$ J(p) = \int_{0}^{\infty} \exp\left( -a x^2 - \frac{p}{x^2} \right) \, \mathrm{d}x. $$
Assume for a moment that $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3307711",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Mathematical music theory concerning melodic intervals and chord progressions There are many books exploring musical theory with maths. However, so far I have only seen discussions about the consonance/dissonance of two notes played simultaneously (intervals) -- this is the theory of "the vertical" on the score. Such t... | Your question is much more related with the realm of perception of notes and classical music theory than mathematics.
For musical note sequences (lines) not having too large an interval between notes: This has to do with perceptual grouping strategies our brain employs. If jumps are too large they are not perceived as... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3307879",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.