Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
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Problem in understanding the partial differentiation I was reading the book on Neural Network and got stuck up on this equation given below.
\begin{eqnarray}
z^{l+1}_k = \sum_j w^{l+1}_{kj} a^l_j +b^{l+1}_k = \sum_j w^{l+1}_{kj} \sigma(z^l_j) +b^{l+1}_k.
\tag{43}\end{eqnarray}
here w and b are constant vectors. The e... | The $z_k$'s are defined over a sum of $z_j$'s.
$$
z^{l+1}_k = \sum_j w^{l+1}_{kj} a^l_j +b^{l+1}_k = \sum_j w^{l+1}_{kj} \sigma(z^l_j) +b^{l+1}_k.
$$
You are differentiating with respect to only one of the $z_j$'s. So, for each $j$, all the other terms in the sum go to zero:
$$
\begin{align}
\frac{\partial z^{l+1}_k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2240044",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Limit of uniformly continuous functions with convergent improper integrals. I am struggling with the proof of the following theorem:
Let $f(x)$ be uniformly continuous in $[a, \infty)$ s.t. the integral $\int_a^{\infty} f(x)dx$ converges. prove that $\lim_{x \to \infty} f(x) = 0$.
I came to the conclusion that it is en... | Hint. If $\lim_{x\to\infty} f(x)$ fails to exist but the integral $\int_{a}^{\infty} f(x) \, dx$ converges, then we often observe a 'train of narrowing peaks':
$\hspace{3em}$
This means that you begin to see an abrupt change in $f(x)$ for large $x$. How the uniform continuity enters this picture is that it prevents pea... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2240140",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Coercive continuous function on a closed subset has a global minimum proof I would like to ask for help with the proof of the following proposition:
Let f be a real continuous function, defined on a closed set $X \subset \mathbb{R}^n$, which is coercive, i.e. for every sequence $\{x_n\}_{n=1}^\infty$ with $||x_n|| \to ... | Let choose any point in $X$, call it $x_0$.
Since $f$ is coercive, then $\exists k>0\mid ||x||\ge k\implies f(x)\ge1+f(x_0)$.
Note: this is a simple way to guarantee that $f(x)>f(x_0)$. It is not a restriction since coercivity allows to find $k$ for any $A$, in particular $A=f(x_0)+1$.
Now $K=X\cap \overline{B(0,k)}$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2240269",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Rolling $2$ dice: NOT using $36$ as the base? I apologize for such a simple question. It has been a while since I took math classes.
When you roll $2$ dice, there are $36$ possibilities. However, there are only $21$ combinations, if order does not matter. Rolling a $(4,2)$ = rolling a $(2,4)$.
Let's say in a game, rol... | Because to get $(1,1)$, both dice must show a $1$. To get a $1$ and a $2$, it could be either $(1,2)$ or $(2,1)$.
Here's another way to look at it ...
The probability of getting $(1,1)$ is
$$\frac{1}{6}\times\frac{1}{6} = \frac{1}{36}$$
Explanation: The dice are independent and each die has probability $\large{\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2240392",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
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Determining a value $k$ that gives a size = 0.05 test. A single observation $X$ from a normal distribution with mean $\mu$ and $\sigma^2$=1 is used to test $$H_0 : \mu = 1 \ \ \ \text{vs} \ \ \ H_1 : \mu \lt 1 $$
using the critical region $C = {{x : x \lt k}}$
Determine the value of k that gives a size 0.05 test.
My at... | You have most of the pieces. Let's put them together: You want to reject $H_0: \mu = 1$ against
$H_a: \mu < 1$ at the 5% level when $X < k.$
Thus, under $H_0$ you want $P(X < k) = 0.05.$
Putting this into a form so that you can use printed standard normal tables, we have
$$P\left(\frac{X - \mu_0}{\sigma} = X - 1 < k - ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2240487",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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How to show that $\mathbb{Z}\left[\frac{1 + \sqrt{5}}{2}\right]$ is finitely generated? We can say that $\mathbb{Z}\left[\cfrac{1 + \sqrt{5}}{2}\right]$ is finitely generated if minimal polynomial of $\cfrac{1 + \sqrt{5}}{2}$ is in $\mathbb{Z}[X]$. After some calculations it can be shown that $f(X) = X^2 - X - 1 \in \... | You've found the minimal polynomial; this gives a very simple way to rewrite the ring:
$$ \mathbb{Z}\left[ \frac{1+\sqrt{5}}{2} \right] \cong \mathbb{Z}[X] / (X^2 - X - 1) $$
so showing the left hand side is finitely generated as an abelian group is the same thing as showing the right hand side is finitely generated as... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2240560",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
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Find all of the elements of a subgroup generated by a $3$-cycle notation Question: Let $H$ be the subgroup of $A_n$ generated by $(123)$. Write down all elements of $H$. Here is my attempt: $H\le \langle 123 \rangle$ $H$=$\langle a \rangle$ where the order of $a$=$n$ where $\langle a \rangle$=$\langle e,a,a^{2},....,a... | Yes, your answer is correct. $H = \langle (123) \rangle = \{e, (123), (132) \}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2240633",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Collinear intersection points of diagonals in a regular heptagon In a regular heptagon, all diagonals are drawn. Then points $A$, $B$, and $C$ in the diagram below are collinear:
(If the vertices of the pentagon are $P_1, \dots, P_7$ in clockwise order, then $A = P_1$, $B$ is the intersection of $P_2 P_5$ and $P_3 P_7... | In your notations, triangles $P_2P_3C$ and $P_7BP_2$ are similar because
$$\angle \, CP_2P_3 = \angle \, P_4P_2P_3 = \alpha = \angle \, P_2P_7P_3 = \angle \, P_2P_7B \,\,\,\text{ and }$$
$$\angle \, P_2CP_3 = 2\, \alpha = \angle \, P_7P_2P_5 = \angle \, P_7P_2B$$ Therefore,
$$\frac{CP_3}{P_2B} = \frac{P_3P_2}{BP_7}$$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2240755",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Does the series $ \sum_{n=1}^{\infty}(-1)^{n}n^{2}e^{\frac{-n^{3}}{3}}$ converge or diverge? Does the series $$\sum_{n=1}^{\infty}(-1)^{n}n^{2}e^{\frac{-n^{3}}{3}}$$ converge or diverge?
I have attempted using the alternating series test, but couldn't find a useful function to use as $b_n$ and no other tests I know see... | The series converges absolutely. This is because we have $$0 \leq \exp\left(-\frac{n^3}{3}\right) \leq \frac{1}{1 + \frac{n^3}{3} + \frac{n^6}{18}}$$
by considering the Taylor expansion and hence
$$0 \leq n^2\exp\left(-\frac{n^3}{3}\right) \leq \frac{n^2}{1 + \frac{n^3}{3} + \frac{n^6}{18}} \leq 18n^{-4}$$
and $$\sum... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2240847",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Examples on product of weakly differentiable functions
*
*Are there two functions $f,g \in L^1_\text{loc}(\mathbb{R}^N)$, $N\ge 1$, such that $f,g$ are weakly differentiable (have all first-order weak partial derivatives in $L^1_\text{loc}(\mathbb{R}^N)$) but $fg$ has no weak partial derivative?
*Are there two funct... | A good place to look for examples is the power functions. This is because the local integrability of these functions (or lack thereof) is easily determined by comparing the power to the dimension. To address your first item, use the fact that $f(x) = |x|^{-p}$ is locally integrable if and only if $p< N$. For such $f$ w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2240934",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What is the symbolic form of "there does not exist a largest natural number " Other students in office hour said this is the correct form
$(\forall x)(\exists y)(y>x)$
{ for all x natural number, there exists y such that y is greater than x }
But
"there does not exist a largest natural number "
$\neg(\exists x)(x\text{... | In English: "there is no natural n, for which all natural k would be smaller than n".
$\lnot \exists n \forall k (k, n) \in \mathbb{N}^2 \land k < n$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2241040",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
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"answer_id": 3
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Taking Limit to infinity when variable is exponent? Example. I am testing $-1/4$ and $1/4$ for
$$\lim_{n \to \infty} \frac{(-1)^n 4^n x^n}{\sqrt{n}}$$
What happens in the numerator that it makes it equal 1?
Like what happens to each term that it everything ends up as 1?
Any help would be greatly super appreciated! Th... | For $x = -1/4$, you have
$$\lim_{n \to \infty} \frac{(-1)^n 4^n (-1/4)^n}{\sqrt{n}}.$$
Since you have a bunch of terms all raised to the same power in the numerator, we can simplify the equation a bit by pulling these terms together inside one pair of parenthesis:
$$\lim_{n \to \infty} \frac{\big[(-1)(4)(-1/4)\big]^n}{... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Proving elements of a polynomial ring are integral over another. I have a quotient polynomial ring $ R = k[X,Y,Z]/ \langle X^2 - Y^3-1, XZ-1 \rangle$ where $k$ is a field and $X,Y,Z$ are variables.
Let $x, y, z $ be the images of $X,Y,Z$ respectively. Fixing $a, b \in k$ and writing $ t = x +ay +bz$, I need to show t... | We have $x^2=y^3+1$ and $zx=1$. We might as well write $1/x$ for $z$.
Then $t=x+ay+b/x$ so
$$ay=t-x-\frac bx.$$
Then
$$a^3x^2=a^3y^3+a^3=\left(t-ax-\frac bx\right)^3+a^3.$$
Multiplying by $x^3$ gives
$$a^3x^5=(tx-ax^2-b)^3+a^3x^3.$$
If $a\ne0$ this equation can be rewritten as
$$a^3x^6+\textrm{ lower terms in }x, t$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2241341",
"timestamp": "2023-03-29T00:00:00",
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Can you apply proof by contrapositive on proof by contradiction In a proof by contradiction, a false statement implies a contradiction. What happens if you take the contrapositive of this? A not-contradiction implies a not-false statement. I.e. a tautology implies a true statement. Is this right?
| Both are forms of indirect proof. As you would expect, a proof by contradiction makes use of a contradiction. A proof by contrapositive does not.
You can use proof by contradiction to infer that a statement $X$ is true by first assuming $X$ is false and then deriving a contradiction of the form $Y \land \neg Y$ or $Y\i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2241578",
"timestamp": "2023-03-29T00:00:00",
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Closed form expressions for harmonic sums It is well known that there are deep connections between harmonic sums (discrete infinite sums that involve generalized harmonic numbers) and poly-logarithms. Bearing this in mind we have calculated the following sum:
\begin{equation}
S_a(t) := \sum\limits_{m=1 \vee a} H_m \cdo... | Even though this is not conceived as an answer to your specific question which conserns the class of functions needed to represent your sum it also contributes to it as it exhibits a broader class than you mentioned.
Also I think it is an interesting result in itself when it comes to closed formes.
Compact closed expr... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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If Y is complete then B(X,Y) is complete I am reading the proof for this theorem and it goes as follows:
To see that $B(X,Y)$ is complete when $Y$ is complete, let $\left\{ T_n \right\}$ be a Cauchy sequence (of operators) in $B(X,Y)$, that is,
$$\|\ T_n - T_m \|\ \to 0 \hspace{.5cm} \text{as} \hspace{.5cm} n,m \to \in... | First let me answer your explicit question.
The fact that the sequence $T_{n}x$ converges to $Tx$ implies that for every $\delta > 0$, there exists some $N \in \mathbb{N}$ such that $\| T_{n}x - Tx \| < \delta$ for all $n > N$.
Now set $\delta = \epsilon \|x\| /2$.
Now for your implicit question, which seems more impo... | {
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"url": "https://math.stackexchange.com/questions/2241816",
"timestamp": "2023-03-29T00:00:00",
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Does $\sum_{n=1}^\infty \frac{1}{\log(e^{n}+e^{-n})}$ converge or diverge? How would I show that the following series converges or diverges?
$$\sum_{n=1}^\infty \frac{1}{\log(e^{n}+e^{-n})}$$
Any help would be appreciated.
| HINT:
$$\log(e^n+e^{-n})=n+\log(1+e^{-2n})\le n+e^{-2n}\le 2n$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2241955",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Intuitive meaning of $E(X^2)$ and $E(X+a)$? I understand conceptually that $E(X)$ is the average expected value of random variable $X$ over multiple trials over a long period of time i.e. the mean. Similarly, I understand for conditionals that $E(X|Y)$ is the average value of $X$ for all cases where $Y$ has already hap... | *
*$E(X+a):$
is a mean value of some new random variable, exactly the $X+a$, which has just shifted ALL values of $X$ by $a$. Why is it happening? Just remember what is a formal definition of some r.v. $X$, it's a function that takes values in $\Omega$ and map them into some subset of $\mathbb{R}$, i.e.: $$X:\Omeg... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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A Prufer code with 2 unknowns Given a labled tree G with the vertices $\lbrace{1,2,3,4,5,6}\rbrace$,
And the Prufer code of G is $(1,x,2,y)$, and $x,y\in \lbrace{1,2,3,4,5,6}\rbrace$. Which of the following is true:
*
*$x,y\in \lbrace{3,4,5,6}\rbrace$ and $x\neq y$
*$x,y\in \lbrace{3,4,5,6}\rbrace$ but it might be ... | The reason Prüfer codes are so useful is that they provide a bijection between the $n^{n-2}$ labeled trees on $n$ vertices and the $n^{n-2}$ sequences of $n-2$ elements from $\{1,2,\dots,n\}$. Every possible sequence of this form gives a unique labeled tree, and every labeled tree has a unique Prüfer code.
In this case... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Regarding recurrences, why do characteristic polynomials work, and why do we look for the roots? I'll use an example recurrence but my question is meant to be generalized.
Let's say we had some recurrence, such as:
$$F(n) = -8F(n-1) + 9F(n-2) + 92F(n-3) - 140F(n-4)$$
where we already know the first few base constants $... | Two explanations.
First, if you have $c_k a_{n + k} + \dotsb + c_0 a_n = 0$, a (not unreasonable, simple) idea is to try $a_n = \alpha^n$. Substituting, you get that it has to be $c_k \alpha^k + \dotsb + c_0 = 0$. Furthermore, the equation is linear, i.e., multiplying a solution by a constant or adding two solutions g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2242305",
"timestamp": "2023-03-29T00:00:00",
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Prove that $\sum\limits_{k=1}^{\infty}\frac{(-1)^k}{2k+1}x^{2k+1}$ uniformly converges on $[-1,1].$ Prove that $\sum\limits_{k=1}^{\infty}\dfrac{(-1)^k}{2k+1}x^{2k+1}$ uniformly converges on $[-1,1].$
My book says I have use alternating series test. I can see that the series converges for any $x\in[-1,1]$ by the altern... | Let $$f_n(x)=\sum_{k=1}^{n} \frac{(-1)^k}{2k+1} x^{2k+1},$$ and call $$f(x)=\sum_{k=1}^{\infty} \frac{(-1)^k}{2k+1} x^{2k+1}.$$
Note that $$f_n'(s)=-s^2\frac{1-(-s^2)^n}{1+s^2} \implies f(x)=- \int_0^x s^2\frac{1-(-s^2)^n}{1+s^2}\,ds$$ and $$f(x)=\arctan(x)-x=-\int_0^x\frac{s^2}{1+s^2}\,ds.$$
Thus:
$$|f_n(x)-f(x)|=\le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2242411",
"timestamp": "2023-03-29T00:00:00",
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Limit of a product of sequence
Let $a > 0$ and $\{y_n\}_{n\geq0}$ be a sequence such that $y_0 > 0$, $y_n > a \ \forall \ n > 1$ and
$$\sum_{n=1}^\infty \frac 1 {y_n} \to \infty$$
Prove That
$$ \lim_{n\to \infty} \prod_{k=1}^n \left( 1 - \frac a {y_k}\right) = 0 $$
I tried to expand the product using Vieta's formula,... | By the strict convexity of the exponential function, we have
$$1 - x \leqslant e^{-x}\tag{1}$$
for all $x\in \mathbb{R}$, and the inequality is strict for all $x \neq 0$. The assumptions yield $0 < 1 - \frac{a}{y_k} \leqslant 1$ for $k > 1$ (probably a typo and it should have been $\geqslant 1$, but that doesn't matter... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2242721",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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An operator is injective if and only if the range of its adjoint separates points Let $T \in L(E,F)$ where $E,F$ are normed spaces. Then, $T$ is injective if and only if $ T^*(F') \subset E' $ separates points of $E$.
$T^*$ means adjoint of T.
I do not have a starting point. I don't even know what separates means in... | It's convenient to use the bracket notation $\langle x,\varphi \rangle =\varphi(x)$ when discussing the spaces and their duals. So, the adjoint $T^*$ satisfies $\langle Tx, \varphi\rangle = \langle x, T^*\varphi\rangle$ for all $x\in E$ and $\varphi\in F^*$. Hence
$$
\|Tx\| = \sup_{\|\varphi\|_{F^*}=1} |\langle Tx, \va... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Show that this function is not square-integrable. Let $f(x):=(1-x^2)^{\frac{m}{2}} \int_0^x \frac{dt}{(1-t^2)^{m+1}}$ be a function on $(-1,1)$. Then I would like to show that the asymptotic of $f$ is such that for $m \in \{1,2,3,...\}$ the function $f$ is not square-integrable at $ \pm 1.$
| You need only look at the case $x=1$. Put $\displaystyle F_m(x)=\int_0^x \frac{dt}{(1-t^2)^{m+1}}$. Then we compute that $F_0(x)=\frac{1}{2}\log (\frac{1+x}{1-x})$. Integrating $F_m$ by parts, we get that
$$(2m+2)F_{m+1}(x)=(2m+1)F_m(x)+\frac{x}{(1-x^2)^{m+1}}$$
Now it is easy to show that $(1-x^2)F_0(x)\to 0$ if $x\to... | {
"language": "en",
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Is $C_c^{\infty}(\mathbb{R}^n)$ dense in $L^p(M,d\sigma)$, $1\leq p<\infty$, where $M$ is an $n-1$ regular surface in $\mathbb{R}^n$? I know that, given an open set $\Omega\subseteq\mathbb{R}^n$, $C_c^{\infty}(\Omega)$ (smooth functions with compact support) is dense in $L^p(\Omega)$, $1\leq p<\infty$.
Let $M$ be a sm... | Let us show that $C_c ^\infty (M)$ is dense in $L^p (M)$ ($1 \le p < \infty$), in two steps.
First, $C_c ^\infty (M)$ is dense in $C_0 (M)$ (the space of functions that vanish at infinity) in the topology of compact convergence, by one of the many variations of the Stone-Weierstrass theorem. Since $C_c(M) \subseteq C_0... | {
"language": "en",
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If $f$ is continuous and $\int^{a}_{-a} f(x) \,dx=0$, is $f$ necessarily odd? I understand that if $f$ is an odd function then
$$\int^{a}_{-a} f(x) \,dx =0$$
Can I say that $f$ is odd function if $\int^{a}_{-a} f(x)=0$ and $f$ is continuous on $[-a,a]$?
| No. Let $a = 1$ and consider the function
$$ f(x) = \begin{cases}\frac{3}{2}x^2&\text{if $x\geq 0$} \\ x&\text{if $x<0$} \end{cases} $$
$f$ is continuous at $0$ since $\lim_{x\to 0^+} f(x) = \lim_{x \to 0^-}$. Furthermore, we have
$$ \int_{0}^1 f(x) = \int_{-1}^0 \frac{3}{2} x^2 = \frac{1}{2} $$
and
$$ \int_{-1}^0 f(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2243348",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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} |
complex number with determinant Let $z_1$ and $z_2$ be two distinct complex numbers and let
$\,z=(1-t)z_1 +tz_2\,$ for some real number $t$ with $0<t<1$. Then we have to prove
$$\begin{vmatrix} z-z_1 & \overline{z}-\overline{z_1} \\
z_2-z_1 & \overline{z_2}-\overline{z_1}
\end{vmatrix}\;=\;0$$
I thought about it, but... | Given that $z=(1-t)z_1+tz_2$ we can write
$z=z_1+t(z_2-z_1)$ so that $z-z_1=t(z_2-z_1)$ and
$$\overline{z}-\overline{z_1}=\overline{z-z_1}=\overline{t(z_2-z_1)}=t(\overline{z_2}-\overline{z_1}),$$
so the top row of the matrix is $t$ times the bottom row. Hence the determinant is zero.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2243476",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find $\sin A$ and $\cos A$ if $\tan A+\sec A=4 $ How to find $\sin A$ and $\cos A$ if
$$\tan A+\sec A=4 ?$$
I tried to find it by $\tan A=\dfrac{\sin A}{\cos A}$ and $\sec A=\dfrac{1}{\cos A}$, therefore
$$\tan A+\sec A=\frac{\sin A+1}{\cos A}=4,$$
which implies
$$\sin A+1=4\cos A.$$
Then what to do?
| Together with
$$\sin A+1=4\cos A $$
you can use $\sin^2A+\cos^2A=1$ as
$$(\sin A+1)(\sin A-1)=\cos^2 A\ .$$
Putting the two together you easily get
$$4\cos A(\sin A-1)=\cos^2 A\ ,$$
and hence
$$4\sin A-4=\cos A\ .$$
You now just have to solve the linear system in $\sin A$ and $\cos A$:
$$\begin{cases}
4\sin A-4=\cos A... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2243571",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 4
} |
Finding the solutions for $\cot(x) = -x$ without digital tools All day I've been stuck with the following question where we're not allowed to use any digital tools. How does the amount of roots depend on the constant $a$?
$\cos(x) = ax$.
*
*$a = \frac{\cos(x)}{x}$,
*$a' = \Big(\frac{1}{x}\Big)\Big(\frac{\cos(x)}{x}... | With the exception of special cases such as $a=0$ or $a=\frac{2}{\pi}$ the equations you are investigating can not be solved using elementary functions (exponents, logarithms, trigonometry, polynomials, etc.). Even using digital tools you still only get an approximate answer.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2243633",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
If $f$ is differentiable everywhere and $|f(y)-f(x)| \leq |x-y|^n$ for all $n >1$, then $f^{\prime}(x)= 0 $ for all $x$. Suppose $f:\mathbb{R} \rightarrow \mathbb{R}$ is a differentiable function and satisfies $|f(x) - f(y)| \leq |x-y|^n$ for all $n > 1.$ Show that $f^{\prime}(x)=0$ for all $x\in \mathbb{R}.$
My first ... | Let's make it more general: if $f:{\mathbb R}\to {\mathbb R}$ is a function satisfying $|f(x) - f(y) |\le |x-y|^{1+\alpha}$ for some $\alpha>0$, then $f$ is constant.
Let's see why. Let $x=0$. For any $n\in{\mathbb N}$, telescoping and triangle inequality give
$$|f(y) - f(0)|\le \sum_{k=1}^n \left |f(\frac{y k}{n})... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2243741",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Given two square matrices $A$ and $B$, is $(AB)^2 = A^2B^2$ true or false? I tried with a counter example and it came out that this claim is false.
I took a matrix
$$
A=
\left[ {\begin{array}{cc}
2 & 1\\
3 & 2\\
\end{array} } \right]
$$
and a matrix
$$
B=
\left[ {\begin{array}{cc}
1 & 3\\
4 & 1\... | Assuming , $A$ and $B$ are invertible, we have $$(AB)^2=A^2B^2$$ if and only if $$ABAB=AABB$$ if and only if $$BA=AB$$ which is not true in general.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2243927",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Compute the limit. Compute the limit. $$\lim _{n \to \infty} \left(\sqrt{n} \int_{0}^{\pi} (\sin x)^n dx\right)$$ I have no clue where to start with this problem so any help is greatly appreciated.
| Another possibility besides the explicit calculation of the integral (which one can do recursively, for example) is to use Laplace's method to approximate
$$
\int_0^\pi\sin^n(x)\,dx=\int_0^\pi e^{n\ln\sin(x)}\,dx=\int_0^\pi e^{n\,f(x)}\,dx
$$
where (in Wiki notations) $f(x)=\ln\sin(x)$ and $x_0=\frac{\pi}{2}$. We have ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2244030",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Proving $C_0^1(\Omega)$ is not dense in $H^1(\Omega)$. Hi I am trying to work on the following problem:
(a) $C_0^1(\Omega)$ is dense in $L_2(\Omega)$
(b) $C_0^1(\Omega)$ is dense in $H_0^1(\Omega)$.
(c) Explain why $C_0^1(\Omega)$ is not dense in $H^1(\Omega)$.
I know how to do (a) and (b) but I couldn't find how to ... | You can check that via looking at the trace. Let $D$ denote the closure of $C_0^1(0,1)$ in $H^1(0,1)$. By definition, the trace of $u \in C_0^1(0,1)$ is zero at both end points. Moreover, it is continuous. Hence, the trace of any function in $D$ is $0$. However, there are functions in $H^1(0,1)$ with non-zero trace, e.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2244142",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
} |
What is the most efficient way to calculate $R^2$? Hello I am working on a question from an old exam paper and wondered what is the best way to tackle parts ii and iii. Given the data it is easy to find $\hat{\beta_0}=-1.071$ and $\hat{\beta_1}=2.741$.
Now for part ii) I have the formula $R^2=1-SSE/SST$ where $SST=\sum... | Yes, you are correct that computing SSE using the definition is a tedious job. If the regression coefficients are estimated without rounding errors, then the following procedure may be used for computing SSE.
By definition,
$SSE = \sum(y_i-\hat{y_i})^2$, and $\hat{y}_i=\hat{\beta}_0 + \hat{\beta}_1 x$.
Plug-in $\hat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2244234",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Conditional probability and indicator function Can someone give me a formal rigorous proof of the following equation?
$$\frac{E\{X \cdot I(T=1) \}}{\Pr(T=1)}= E(X|T=1)$$
Many thanks!
| \begin{align}
\mathsf{E}(X \cdot \mathbb{I}(T = 1)) &= \sum_t \mathsf{E}(X \cdot \mathbb{I}(T = 1) \mid T = t)\cdot\Pr(T = t) \\
&= \mathsf{E}(X \cdot \mathbb{I}(T = 1) \mid T = 1) \cdot \Pr(T = 1) \\
&= \mathsf{E}(X \mid T = 1) \cdot \Pr(T = 1)
\end{align}
where the first equality is by the law of total expectation.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Verifying that the $2n$-th term in a modified generalized Fibonacci sequence of order $n$ is the $n-1$-th Cullen number I was messing around with recursive sequences and I came across something interesting.
The Fibonacci numbers start with $F_1 = 0$, $F_2=1$, and continues with $F_i = F_{i-1} + F_{i-2}$.
The Tribonacci... | Prove that $G_{n+k}=2^{k-1}(n-1)+1$ for $1\le k\le n$.
It's useful to note that $G_{n+k}=2G_{n+k-1}-G_{k-1}$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Probability of choosing same set of unique objects over time without replacement I have $1024$ unique objects, and each round I start with all of them and choose $7$ uniformly at random without replacement. If I do this $n$ times, what is the probability distribution that I will have a collision (pick the same $7$ as I... | The number of different choices of $7$ of the $1024$ objects is $\binom{1024}7$. This is a very large number; call it $C$.
If $n=2$, there is a collision with probability $\frac 1C$ and so no collision with probability $\frac{C-1}C$. If $n=3$, to avoid a collision we first have to choose differently in the first two r... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Vorticity equations of incompressible Navier-Stokes equations in 2D We know for incompressible Navier-Stokes equations, we have the vorticity equation: $$\omega_t - \Delta \omega + (u \cdot \nabla)\omega = (\omega \cdot \nabla)u$$
But for two dimensional space, $(u \cdot \nabla)\omega $. I don't see why after I plug in... | A solution to the 2D Navier-Stokes equations can be realised as a special kind of solution to 3D Navier-Stokes, namely one for which
$$ u(x,y,z) = (u_1(x,y),u_2(x,y),0) , \quad w(x,y,z)=\nabla \times u(x,y,z) =(0,0,\omega(x,y)).$$
Here, $\omega = \operatorname{curl}_{\text{2D}}(u_1,u_2). $ See for instance here.
Pluggi... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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} |
Finding eigenvectors of a 3x3 matrix with a root of multiplicity 3 I have the matrix \begin{bmatrix}1&0&0\\2&2&-1\\0&1&0\end{bmatrix}
I know that the only eigenvalue is 1 with multiplicity 3
I solved for the first eigenvalue and got \begin{bmatrix}0\\1\\1\end{bmatrix}
How do I find the other two? I know they are \begi... | If you use a Linear Algebra tool like in MATLAB, or a libraries for programming languages, like NumPy with Python or MathNet Numerics with C#, you can find out that the eigenvectors calculated by these tools are:
[0, 0, 0]
[(√2)/2, (√2)/2, (√2)/2]
[(√2)/2, (√2)/2, (√2)/2]
The eigenvalues are as stated:
(1,0)
(1,0)
(1,0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2244911",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
} |
Finding Coordinates on a 3D vector My question is how to find the coordinates for a point on segment $KL$ where
$K=(3,2,1)$ and $L=(7,9,5)$ that is $5$ units away from $K$.
I know that the vector is $KL=[4,7,4]$ and the length of the entire vector is $9$ but not sure how to move the point along the vector by $5$ units... | We want to move $\dfrac 59$ of the vector $[4,7,4]$, which is $\left[2\frac29, 3\frac89,2\frac29\right]$
This equates to the point
\begin{align}P&=K+\frac 59(KL)\\
&=(3,2,1)+\frac59[4,7,4]\\
&=(3,2,1)+\left[2\frac29, 3\frac89,2\frac29\right]\\
&=\left(5\frac 29,5\frac89,3\frac29\right)\\
&\approx (5.22, 5.89, 3.22)\qu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2245009",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proving $\lim_{(x,y)\to (0,0)} (x^4+y^4)/(x^2+y^2)=0$ by definition I need to prove by definition (and nothing else) that
$$\lim_{(x,y) \to (0,0)}\frac{x^4+y^4}{x^2+y^2} = 0.$$
I've been stuck on this for almost an hour with no luck, and ran out of ideas.
Can anyone help or give a hint?
| Another approach which is useful whenever you see a denominator of $x^2+y^2$ is to use polars: $x=r\cos t$, $y=r\sin t$. Then
$$\frac{x^4+y^4}{x^2+y^2}=\frac{r^4\cos^4t+r^4\sin^4t}
{r^2\cos^2t+r^2\sin^2t}=r^2(\sin^4t+\cos^4t)$$
which is clearly $\le 2r^2$. So as $r\to0$ the function tends to zero.
(Although as Luiz poi... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 0
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What is the standard deviation of dice rolling? When trying to find how to simulate rolling a variable amount of dice with a variable but unique number of sides, I read that the mean is $\dfrac{sides+1}{2}$, and that the standard deviation is $\sqrt{\dfrac{quantity\times(sides^2-1)}{12}}$.
I doubt that the $12$ comes ... | The formula is correct. The 12 comes from $$\sum_{k=1}^n \frac1{n} \left(k - \frac{n+1}2\right)^2 = \frac1{12} (n^2-1) $$
Where $\frac{n+1}2$ is the mean and k goes over the possible outcomes (result of a roll can be from 1 to number of faces, $n$), each with probability $\frac1{n}$. This formula is the definition of v... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
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Lambert W function style inequality Suppose that for real $x,y \ge 1$ we have the inequality $x/\log x \le y/\log 2$.
Then the inequality
$$
x \le (2/\log 2)y\log y + c
$$
appears to hold for some constant $c$.
What is the smallest such $c$?
I would also accept an argument that no such constant exists.
Depending on h... | $x/\log(x)$ for $x > 1$ has a minimum value of $e$ at $x=e$, decreasing for $1 < x < e$ and increasing for $e < x < \infty$. Thus if $y < e \log(2)$ there are no solutions to your inequality; if $y \ge e \log(2)$ the solutions form an interval
$$ - \frac{y W_{0}(-\log(2)/y)}{\log(2)} \le x \le - \frac{y W_{-1}(-\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2245321",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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order of the pole $0$ of function $f(z)=\frac{1}{z^{3/2}}$
Find order of the pole $0$ of $f(z)=\frac{1}{z^{3/2}}$.
I think it is $2$ because
\begin{align*}
z^{3/2}\big|_{0}&=0\\
(z^{3/2})'\big|_{0}&=0\\
(z^{3/2})''\big|_{0}&\neq 0,
\end{align*}
but I'm not sure. Could someone explain to me, please?
| As Chappers pointed out this function is not analytic on any annalus centred at $z=0$. It's a multivalued function defined by:
\begin{align*}
z^{-\frac32} = e^{Ln(z^{-\frac32})} = e^{-\frac 32 (\ln|z| +i\arg(z)+2p\pi i)} = |z|^{-\frac32} e^{\frac{-3i}{2}(\arg(z)+2p\pi)},
\end{align*}
where $p \in \mathbb Z$.
Another ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Quantifier statement explanation
I'm having trouble understanding what the above statement means in plain English.
Does it mean: "For any Natural number there exist a number greater than it". I don't get what the right part of the implication means.
| You are correct. Literally translated, the statement means "for all $x$ such that $x$ is a natural number, there exists a number $y$ such that $y$ is a natural number and $x$ is less than $y$".
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2245525",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Sum of squares by treating it as a nonhomogenous recurrence Let $T(n) = T(n-1) + n^2$ where $T(0) = 0$.
The homogenous part $T(n) = T(n-1)$ has characteristic polynomial $x - 1 = 0$ and root $1$, which means $T(n) = \alpha \cdot 1^n$ for the homogenous part.
I am not sure how to do the nonhomogenous part. I tried this:... | You should try with a full cubic, not just the highest degree term. So you have to try $T(n) = \alpha n^3 + \beta n^2 + \gamma n + \delta$. Setting $T(0)=0$ we have $\delta=0$.
Then, setting $T(n) = T(n-1) + n^2$ yields:
$$\begin{cases} -3\alpha -1 = 0 \\ 3\alpha - 2\beta = 0 \\ -\alpha + \beta -
\gamma = 0\end{cases... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2245608",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Set Proof verification
How to prove that $(A-B)\cup(A\cap B)=A$?
Here is my proof, I'm not sure if I verified it correctly.
$(A-B) \cup (A \cap B)$ simplifies into $x \in (A-B) \vee x \in (A \cap B)$.
Then we have $[x \in A$ and $x \notin B$] or $[x \in A$ and $x \in B$] $\iff x \in A$
Therefore, $(A-B) \cup (A \ca... | Provide another proof:
Notice that $(A-B) \subset A$ and $(A\cap B) \subset A$, thus $(A-B) \cup (A\cap B) \subset A$
On the other hand, $\forall x \in A$, if $x\in B$, then $x\in A\cap B$; else if $x\notin B$, then $x\in A-B$. Thus $x\in (A-B) \cup (A\cap B)$. So we have $A\subset (A-B) \cup (A\cap B)$.
So we get $A =... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proof that $\sin$ is continuous by open sets on $\mathbb{R}$ with the usual topology The open set definition of continuity is:
$f:A \to B$ is continuous $\iff$ $U_{B}\in\tau_{B} \implies f^{-1}U\in\tau_{A}\ \forall U_B$, where $\tau_A$ and $\tau_B$ are the topologies of $A$ and $B$.
I believe that in the usual topology... | If you want to use the "inverse image of open sets are open" definition of continuity you could begin by noting that there are three types of basic open intervals in $[-1,1]$
*
*$(a,b)$ where $-1<a<b<1$
*$(a,1]$
*$[-1,b)$
Then point out the the inverse image under $\arcsin$ of each of these three types of basic op... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Information Encoded by a Probability Density Function I want to calculate the information needed to encode a probability density function. For a discrete probability function such as a coin flip, the information would be calculated as follows:
$$S=\sum_n -P_n\log_2P_n$$
So for a coin flip we would have
$$S=-0.5\log_20.... | The amount of information (in the Shannon sense, measured in bits) that a continuous source produces is infinite. You need an infinite amount of bits to encode a variable that -for example- is uniform on $[0,1]$. The differential entropy is not the same as the true entropy, it cannot be interpreted in that way.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2245876",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Sum of Idempotent Transformation
Let $V$ be a $n$ dimensional linear space on the number field $K$. Let $A_1,A_2,\cdots,A_s$ be idempotent transformations(or matrices) on $V$ ($A_i^2=A_i$, $i=1,2,\cdots,s$). If $A=A_1+A_2+\cdots+A_s$ is also an idempotent transformation, Prove $A_iA_j=0$ and $A_jA_i=0$ for $1\leqslant... | Answer inspired by the one given at the link in the comment under the question.
Note that a linear operator being idempotent means that it is diagonalisable with eigenvalues $0$ and $1$ only: the space $V$ is a direct sum of the eigenspaces for $0$ (its kernel) and the eigenspace for$~1$ (its image).
Among other thing... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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Do I have to multiply it whole? Given that $$A=\begin{bmatrix}4 & 1\\ -9 & -2 \end{bmatrix}$$
and $$A^{100}=\begin{bmatrix}a & b\\ c & d \end{bmatrix}$$
What is $a$?
I tried to multiply it again and again but it seems lengthy. Is there a shorter method?
| From
$$
A^{k+1} = A^k A
$$
you have the recurrences
$$
a_1 = 4 \\
b_1 = 1 \\
a_{k+1} = 4 a_k - 9 b_k \\
b_{k+1} = a_k - 2 b_k
$$
for the upper elements of $A^{k+1}$.
Then
$$
2 a_{k+1} = 8 a_k - 18 b_k = - a_k + 9 b_{k+1} \\
9 b_{k+1} = 2 a_{k+1} + a_k
$$
and
$$
a_{k+2}
= 4 a_{k+1} - 9 b_{k+1}
= 2 a_{k+1} - a_k
$$
whic... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2246077",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
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Help me understand division in modular arithmetic From Wikipedia:
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value
So the point of modular arithmetic is to do our normal arithmetic operations wrap around after reaching a certain value.
... | I'll try my best to answer concisely:
*
*It is not a condition for $a,b$ to be in $[o,n)$ but since every integer is equivalent to one, it makes sense to use numbers in this range
*When I studied this, I liked to think that rather than division of $a$ by $b$, we multiply by the inverse of $b$. If $a|n$ there is no ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2246175",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
What is the average? When I was first introduced to the concept of average (mean), I was confused. What does average mean? How does one number $\sum_{i=1}^{n} a_{i}$ represent the "central tendency" of a set of data points $a_i$. Then I found a way to deal with this concept. I thought that the average (mean) is "the cl... | Your attempts is excellent, but will not lead you to success.
Actually, the minimum of
$$f_1(x) := \sum_{i=1}^{n}|x-a_i|$$
is known to be achieved by the median, another estimator of the central trend.
(The median is defined as the value of rank $n/2$ or the average of the two values of rank $(n\pm1)/2$; for even $n$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2246270",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
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For what value of $w$ is $(1-w)\bar X_1 + w\bar X_2$ the minimum variance unbiased estimator of $\mu$ Let $\bar X_1$ and $\bar X_2$ be the means of two independent samples of sizes $n$ and $2n$ from an infinite population that has mean $\mu$ and variance $\sigma^2 \gt 0$. For what value of $w$ is $(1-w)\bar X_1 + w\bar... | For any distribution, $E(X_i) = \mu$. If the population mean is $\mu$, any single observation $X_i$ has mean $\mu$.
In addition, if $\{X_i\}$ is an iid sample, then the expectation of the sample mean is $\mu$ (i.e., sample mean is an unbiased estimator of $\mu$). Because of that, for any $w$ the estimator $(1-w) \bar{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2246335",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Can Fermat's Two Squares Theorem be phrased in terms of Schemes? Fermat's two squares theorem says that a prime number $p = a^2 + b^2$ is the sum of two squares if and only if $p = 4k+1$. How might I phrase this in terms of Schemes?
I know that $\mathrm{Spec}(\mathbb{Z}) = \{ primes \}$. And maybe we are saying there ... | We have that $$p=x^2+y^2$$defines a conic over $\mathbb{Q}$ and its rational points are solutions to this equation in $\mathbb{Q}$. Here we always add the points at infinity to make it a projective curve. But the same equation defines a conic bundle scheme over $\mathbb{Z}$ whose generic fiber is that conic over $\math... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2246415",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 1,
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Solve $(y+\sqrt{x^2+y^2})dx-xdy=0$ Solve $(y+\sqrt{x^2+y^2})dx-xdy=0$
I suspect this is homogoneus equation after we divide sides by $y$. But I don't know how to contiunue.
| note that we can write $$1+\sqrt{\left(\frac{x}{y}\right)^2+1}-\frac{x}{y}\frac{dy}{dx}=0$$ for $$y>0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2246649",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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How do we conclude that following two groups are isomorphic? Let $H= \Bbb Z \times \Bbb Z \times \{0\}$ and $T = \{0\} \times \Bbb Z \times \{0\}$
Then $H/T $ is isomorphic to $\Bbb Z$
$\Bbb Z$ denotes integers, and $\times$ Cartesian product.
How did we make that conclusion? I come across such examples very often, co... | Define a map $\phi:H \to \mathbb{Z}$ such that $(m,n,0) \mapsto m$. It is easily verified that this is a group homomorphism, and that the image of $\phi$ is all of $\mathbb{Z}$. Moreover note that $\ker{\phi} = T$. Therefore, by the First Isomorphism theorem we have that:
$$
H/T \cong \text{Im}{\phi} = \mathbb{Z}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2246770",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Find area of trapezoid contained in circle
The picture below represents a circumference of radius 5 and center at
the origin of the referencial 0xy. In the picture is represented too a
trapezoid [ABCD], with [AD] and [BC] parallel. A is the point of the
circumference that belongs to the x axis. Points B and C be... | You were correct but you did not continue with the first formula you derived: $$ {2⋅5cos\alpha+(5+5/2)\over 2}⋅5sin\alpha = 5{10\over 2}cos\alpha sin\alpha + 5{7.5\over 2}sin\alpha ⇒ \\ A(\alpha) = 25sin\alpha cos\alpha +{75\over 4}sin\alpha$$
Here's an alternative way:
We have to notice that as the angle a changes the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2246831",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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} |
Find three subspaces $W_1$, $W_2$ and $W_3$ of $k [X]$ such that $k[X]\cong W_1\oplus W_2\oplus W_3$ I am trying to find three non-trivial subspaces of all polynomials, there are infinite subspaces of the space of all polynomials? I could fix a natural $n$ and say that the polynomials of degree less than $n$ are a subs... | A basis of $k[X]$ is $1, X, X^2, X^3, \dots$. The only thing you have to do is divide the basis elements in three groups, so for instance
$$\begin{align*}
W_1 &= \text{span}(1) = k \\
W_2 & = \text{span}(X) = k \cdot X \\
W_3 & = \text{span}(X^2, X^3, \dots) = X^2 \cdot k[X]\\
\end{align*}
$$
would work.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2246977",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Can any graph be represented by a formula? I am only in high school but interested in maths. Can any graph be represented by a formula or does the graph have to have certain characteristics like a pattern etc?
| If you're asking whether a function has to be given by some sort of algorithm or logic, then the answer is no: A function $f:X \to Y$ is simply a subset of $S_f \subset X\times Y$ such that for each $x\in X$, there exists exactly one point $(x, y)\in S_f$; the point $y$ is denoted by $f(x)$. Note that this assignment i... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to solve this logarithmic equation through algebraic manipulation? I was working through an Art of Problem Solving workbook when I encountered a very frustrating problem.
Solve the equation $\log_{2x}216=x$, where $x\in \mathbb{R}$.
I understand how to find the answer by inspection (all you have to do is stare at i... | In terms of using inspection, I think this makes the inspection easier.
$$\ln_{2x}216=x \Rightarrow \ln 6^3 = x \ln 2x \Rightarrow 3 \ln 6 = x \ln2x $$
If you guess that the linear factors are equal ($x=3$) you immediately see the logarithmic factors are equal.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2247182",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
The number of solutions of the equation $\tan x +\sec x =2\cos x$ lying in the interval $[0, 2\pi]$ is: The number of solutions of the equation $\tan x +\sec x =2\cos x$ lying in the interval $[0, 2\pi]$ is:
$a$. $0$
$b$. $1$
$c$. $2$
$d$. $3$
My Attempt:
$$\tan x +\sec x=2\cos x$$
$$\dfrac {\sin x}{\cos x}+\dfrac {1}{... | The answer must be 2
Since tanx and secx domain would consider every x, rejecting (2n+1)π/2 values in the interval [0,2π] such values would be rejected from the solution.
In the last step sinx=1 the solution would be 3π/2 and this would be rejected due to domain condition. The other part, that is, sinx=1/2 has solutio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2247408",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why are numerator and denominator called so? There are terminologies for natural numbers, whole numbers and so on. (If the meaning of the terms can be found, it becomes easier to understand. For natural numbers, the term "natural" refers to the naturally occurring set of numbers in nature like $2,3,4$ and not $-2$, $-3... | In a fraction, such as two-fifths, "two" is the numerator, and "fifths" the denominator.
Numerator tells us "how many". The word is derived from the Latin "numerus" (number).
Denominator names the "things" we are counting. The word is derived from the Latin "denomino" (to name).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2247583",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
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Convergence Speed for Optimization Methods on non Lipschitz and strictly monotone funtions I am studying convergence analysis for some optimization techniques, so this could be a naive question. In the derivation of convergence speed for Gradient descend (and Newton method), they usually assume Lipschitz condition on t... | Perhaps what you're looking for is the subgradient descent algorithm, which works without assuming that the function $f$ being optimized is even differentiable; one only needs continuity, and a Lipschitz condition on $f$ itself, i.e.
$$ |f(x)-f(y)|\leq L\|x-y\|.$$
Of course, we also need access to the subgradient infor... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2247681",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Suppose equation $x^{12} = 1$ has $14$ solution in some group. Show that this group is not cyclic.
Suppose equation $x^{12} = 1$ has $14$ solution in some cyclic group.
Show that this group is not cyclic.
Any help would be appreciated. I was trying to show this by contradiction, but I didn't go too far.
Attempt:
Su... | Here is another take.
There is only one finite cyclic group of each order, up to isomorphism.
$\mathbb C^\times$ contains a copy of the cyclic group of order $n$: it is the subgroup of $n$-th roots of unit.
The equation $x^{12} = 1$ has at most $12$ solutions in $\mathbb C$ and so cannot have $14$ solutions.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2247847",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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Knowing that for any set of real numbers $x,y,z$, such that $x+y+z = 1$ the inequality $x^2+y^2+z^2 \ge \frac{1}{3}$ holds.
Knowing that for any set of real numbers $x,y,z$, such that $x+y+z = 1$ the inequality $x^2+y^2+z^2 \ge \frac{1}{3}$ holds.
I spent a lot of time trying to solve this and, having consulted some ... | Here's another way to approach this. It's easy to see that the value of $\frac13$ is obtained when each of $x, y, z$ is $\frac13$. We want to show that as the variables deviate from this point (with their sum still being 1) the value cannot decrease.
So we look at the deviations from $\frac13$: $x=\frac13+\epsilon_1$, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2247973",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 15,
"answer_id": 10
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Why is $\lim_{x \rightarrow e}\frac{\ln{(x)}-1}{x-e}$ equal to $\frac{1}{e}$ As it says. Why? The best I could achieve is:
$$\lim_{x \rightarrow e}\frac{\ln(x)-\ln (e)}{x-e}$$
And the answer says it is equal to
$$\frac{\text{d}}{\text{d}y}(\ln(x))$$
When $x$ is $e$ so it should be
$$\frac 1 x$$
And when $x$ is replace... | The derivative of a real valued function $f$ at a point $y$ is given by $\lim_{x \to y} \frac{f(x) - f(y)}{x-y} = f'(y)$. In your case, if we have $f(x) = \ln x$ then we know $f'(x) = \frac{1}{x}$ and $f'(e) = \frac{1}{e}$.
But $\displaystyle f'(e) = \lim_{x \to e} \frac{\ln x - \ln e}{x-e} = \lim_{x\to e} \frac{\ln x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2248211",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Difficult limit $\lim_{x\to\infty} e^{-x}\int_0^x \int_0^x \frac{e^u-e^v}{u-v} \,\mathrm{d}u\mathrm{d}v$ I need to calculate this limit:
$$\lim_{x\to\infty} e^{-x}\int_0^x \int_0^x \frac{e^u-e^v}{u-v} \,\mathrm{d}u\mathrm{d}v.$$
How do I do it?
There's a hint that I should use de l'Hospital's rule.
| Expand the exponentials in their standard Taylor series to see the integrand equals
$$\sum_{n=0}^{\infty}\frac{1}{n!}\frac{u^n-v^n}{u-v}= \sum_{n=1}^{\infty}\frac{1}{n!}(u^{n-1} + u^{n-2}v + \cdots + uv^{n-2} + v^{n-1}).$$
For each $n$ we have
$$\int_0^x\int_0^x (u^{n-1} + u^{n-2}v + \cdots + uv^{n-2} + v^{n-1})\,dv\,d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2248355",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
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Rewriting a set as a polyhedron I have the following set:
\begin{align}
M = \{ x \in \mathbb{R}^n: x \geq 0, x^{T}y \leq 1, \forall y \text{ with } \lVert y \rVert \leq 1 \}
\end{align}
I would like to rewrite this set so as to find out whether it is a polyhedron, defined as the intersection of finitely many halfspaces... | If the norm is Euclidean then the answer is straightforward.
Note that $\langle x , y \rangle \le 1$ for all $y$ of unit norm or less
iff
$\max_{\|y\| \le 1} \langle x , y \rangle \le 1$
iff $\|x\| \le 1$.
Hence the set in question can be written as
$\{ x | x \ge 0, \|x \| \le 1 \}$.
This is not a polyhedral set.
Note... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2248479",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Computing $\sum_{n=1}^{\infty} \frac{1}{n^4}$ I came across this problem in Fourier analysis, but I'm not sure my confusion is with Fourier analysis per se, it might be that I'm lacking some understanding of sums/series.
I am trying to determine the sum $$\sum_{n=1}^{\infty} \frac{1}{n^4}.$$ I know this problem has bee... |
Although the OP is seeking a way forward through Fourier series analysis, I thought it might be instructive and useful for some users to present an approach that relies on contour integration. To that end, we proceed.
We begin by noting that the function $f(z)=\frac{\cot(\pi z)}{z^4}$ has simple poles at $z=n$, $n\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2248607",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 2
} |
How many free variables can there be for an $n\times n$ matrix when solving for an eigenspace? let's say you have an nxn matrix, call it $A$. If $A =$
$$
\begin{matrix}
a^1_1 & a^1_2 & \cdots & \cdots & \cdots & a^1_n \\
a^2_1 & a^2_2 & \cdots & \cdots & \cdots & a^2_n \\
\vdots & \vdots & \ddots &... | Expanding on my comment above: I think the answer you're looking for is the following: The number of free variables in $\det(A - \lambda I) = 0$ is equal to the dimension of the null space of $A - \lambda I$.
Here are some examples to play with:
$$
A_3 = \begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{pmatrix... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2248707",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Prove that $a_n \to 1$ implies $a_n^2 \to 1$ using the definition of convergence Suppose that $(a_{n})$$_{n \in \mathbb{N}}$ is convergent, with limit 1. Show, directly from the definition, that $(a^2_{n})$$_{n \in \mathbb{N}}$ is convergent, with limit 1.
My Attempt:
Let $\epsilon > 0$ be given. We want to find $N \in... | We know $a_n$ converges to $1$. We know that for each $\delta>0$, there's an $N$ such that for all $n>N$, we must have
$$|a_n-1|<\delta$$
Let $\epsilon>0$. Take $\delta=-1+\sqrt{1+\epsilon}$ (and see $\delta>0$).
Now note that
$$|a_n+1|\leq |a_n|+|1|=|a_n|-|1|+2\leq |a_n-1|+2<2+\delta$$
so that
$$|a_n^2-1|=|a_n-1||a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2248778",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Can a function $f$ have $L^p$ norms $\Vert f\Vert_p =p$ for all $1\leq p<\infty$? I have tried to show that such a function must be in $L^{\infty}$ and thus it is impossible for such a function to exist since, in that case, $$\infty =\lim_{p\rightarrow \infty} \Vert f\Vert_p=\Vert f\Vert_{\infty}.$$
Basic estimates see... | The magic words are: Riesz-Thorin interpolation Theorem.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2248989",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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What is this *redundant cycle* "thing"? Why is it a matroid? I found the following definition in an introductory exposition about matroids.
Let $G=(V,E)$ be a given (finite) graph. Let $\mathcal{C, D}$ be arbitrary
collections of cycles in $G$.
$\mathcal{C}$ is redundant, if every edge in $G$ appears in an even
n... | Suppose you want to count the cycles in a ladder graph.
Clearly it looks as though a ladder graph with $n$ rungs is made of $n-1$ cycles glued together. However, if you count all possible cycles, you find $\binom n2$ of them (choose any two rungs and take the rectangle between them). The notion of independent cycles f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2249115",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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why$2 (-1 + 2^{1 + n})$ is the answer to the recurrence relation $a_{n}=2a_{n-1}+2$? $a_{0}=2$
$a_{1}=2(2)+2$
$a_{2}=2(2(2)+2)+2$
$a_{3}=2(2(2(2)+2)+2)+2$
$a_{4}=2(2(2(2(2)+2)+2)+2)+2$
$a_{5}=2(2(2(2(2(2)+2)+2)+2)+2)+2$
To simplifiy
$a_{6}=2^{6}+2^{5}...2^{1}$
so my answer is
$a_{n}=2^{n+1}+2^{n}+...2^{1}$
The correct... | The inhomogeneous recurrence relation
$$
a_n = 2 a_{n-1} + 2
$$
can be turned into a homogeneous recurrence
$$
a_n - a_{n-1} = 2 a_{n-1} + 2 - (2 a_{n-2} + 2) = 2 a_{n-1} - 2 a_{n-2} \iff \\
a_n = 3 a_{n-1} - 2 a_{n-2}
$$
and solved by the usual algorithm.
The characteristic polynomial is
$$
p(t) = t^2 - 3 t + 2
$$
wit... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2249216",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 1
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Gradient computation, result verification I have a problem with the computation of the gradient of the function
$$
L(w) = -\dfrac{1}{N}\sum\limits_{n=1}^N y_{n}\log\left( \sigma(w^{T}x_{n}) \right) + (1 - y_{n})\log\left( 1-{\sigma}(w^{T}x_{n}) \right)
$$
where $\sigma$ is a sigmoid function defined by $\sigma(x) = \df... | An easy way to check is to check the components.
Recall that
$$
\frac{\partial}{\partial x}\sigma(x) = \sigma(x)[1-\sigma(x)]
\;\;\;\&\;\;\;
\frac{\partial}{\partial w_i} w^Tx_j = x_{ji}
$$
Then, the $i$th component of the gradient is:
\begin{align}
\frac{\partial}{\partial w_i} L(w)
&=
\frac{-1}{N}\sum_j y_j\frac{\sig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2249351",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Uniform convergence of $\sum\limits_{n=1}^{\infty}(1-x)x^n$ I want to study the uniform convergence of $\sum\limits_{n=1}^{\infty}(1-x)x^n \rm{~~for}~~ x$ on $[0, 1]$. This is my attempt:
First I study convergence on $[0, 1)$: by Ratio test sum is convergent when $|x|< 1$. Hence, $\sum\limits_{n=1}^{\infty}(1-x)x^n$ o... | Hint: $(1-x)x^n=x^n-x^{n+1}$. Just use the partial sums, then everything gets simple as these are partial sums of the geometric series.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2249502",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
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Polygon area method I saw this problem in a puzzle book.
Just wondering if anyone can explain the principle behind this method.
A rectilinear figure of any number of sides can be reduced to a triangle of equal area, and as $\angle AGF$ happens to be a right-angle the thing is quite easy in this way:
*
*Continue the ... | It is the standard Euclidean procedure for reducing any polygon to a triangle.
Take triangle ABC (yellow) and triangle A1C (pink) as in the diagram below:
Since 1B and AC are parallel, the two triangles have the same area. Hence the polygon GABC... has the same area as G1C..., which is a polygon with one less vertex. ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "5",
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Does the geodesic on a surface $z = f(x,y)$ always trace out a straight line in the $xy$ plane? Let $z = f(x,y)$ be a surface. Let $(x_0, y_0, z_0)$ and $(x_1, y_1, z_1)$ be two points on that surface. Let $g(t) = \langle x(t), y(t), z(t)\rangle$ be a parameterization of the geodesic curve between the two points. Is t... | If the rigid surface containing the geodesic curve rolls in the plane of osculation with common normal without slipping then we have the same zero geodesic curvature in the plane as well as the surface, the trace on the plane is a straight line.
The instantaneous projection without any rolling is not in general a str... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 2
} |
Number of Solutions to $e^{z}-3z-1=0$ in the Unit Disk I am working through some of the past qualifying exams in complex analysis and I am a bit stuck on the question I posed in the title. My immediately thought is use Rouche's Theorem. For instance, I tried letting $f(z)=e^{z}$ and $g(z)=3z+1$ in hopes of getting $|f(... | For $z\ne 0$, let $\gamma$ be the line segment connecting $0$ and $z$. Then,
$$|e^z -1| = \left| \int _\gamma e^u du \right| \le \int _\gamma \sup_{u\in \gamma}|e^u| |du| \le e|z| $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2249855",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Discrete math: forming a comittee from n men, n women, using 2 different approaches. Prove that,
$$\sum_{k=1}^n k\binom{n}{k}^2 = n\binom{2n-1}{n-1}$$
by determining, in two different ways, the number of ways a committee can
be chosen from a group of $n$ men and $n$ women. Such a committee has a
woman as the chair and ... | Hint: $k$ is the total number of women on the committee (so must be at least $1$). It may help to use the fact that $\binom nk=\binom n{n-k}$ to rewrite the LHS as $$\sum_{k=1}^nk\binom nk\binom n{n-k}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2249940",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
How to prove this integral $\iint_{D} \frac{\mathrm{d}\bar{z}\mathrm{d}z}{z-\zeta} = - 2{\pi}i{\bar{\zeta}} $ I am reading this paper and there is an integral in it:
$$\iint_{D} \frac{\mathrm{d}\bar{z}\mathrm{d}z}{z-\zeta} = - 2{\pi}i{\bar{\zeta}},$$ where $D$ is a disc of radius $R$ and $\zeta$ is a point in $D$.
I ... | The integral you have isn't $-2\pi i \bar{\zeta}$ unless the disk $D$ is centered at origin.
Assume $D$ is centered at origin and $r$ is its radius. Using Stoke's theorem for complex coordinates, we have
$$\int_D \frac{d\bar{z} \wedge dz}{z-\zeta}
= \int_D d\left( \frac{\bar{z}-\bar{\zeta}}{z-\zeta} dz \right)
= \int... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2250037",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Determine whether $\int^{1/2}_0\frac{1}{\sin x\cdot \ln x}dx$ is convergence, absolute convergence or divergent. I'm trying to determine whether $\int^{1/2}_0\frac{1}{\sin x\cdot \ln x}dx$ is convergence, absolute convergence or divergent.
Let $f(x) = \frac{1}{\sin x\cdot \ln x}$, $f(x)< 0$ for $(0,0.5]$.
Therefore I ... | $\sin{x}<x$ for $0<x<\pi/2$ (draw a picture to see this). So
$$ \frac{1}{\sin{x} \cdot (-\log{x})} >\frac{1}{x \cdot (-\log{x})}, $$
and this has antiderivative $-\log{(-\log{x})}$. This is finite when $x=1/2$ and diverges for $x \to 0$, so the original integral diverges.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2250259",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Placing m books on n shelves such that there is at least one book on each shelf Given $m \ge n \ge 1$, how many ways are there to place m books on n shelves, such that there is at least one book on each shelf?
Placing the books on the shelves means that:
• we specify for each book the shelf on which this book is plac... | Given that the ordering of the books is important, not merely which shelf they are on, you can simply split any book ordering into shelves by choosing the $n{-}1$ shelf breaks from the $m{-}1$ book gaps. So there are
$$m!\binom{m-1}{n-1} = \frac{m!(m-1)!}{(m-n)!(n-1)!} \text{ options}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2250353",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Behavior of $\sin x/x$ as $x$ approaches 0? Which is the limiting behavior, $x \to 0\ \frac{\sin x}{x}$, in terms of x:
1) $\lim\limits_{x \to 0} \frac{\sin x}{x}$ = $\lim\limits_{x \to 0}\frac{\sin'x}{x'} = \frac{\lim\limits_{x \to 0}\cos x}{1} \rightarrow 1 - x^2/2 + O(x^4)$ by L'Hospitals' rule, or,
2) As $x \to 0,\... | Because L'Hopital's theorem says:
Let $f,g$ be two differentiable functions such that $g'(x)\ne 0$ in a neighbourhood of $x_0$, $\lim_{x\to x_0}\frac{f'(x)}{g'(x)}=L\in\Bbb R\cup\{-\infty,\infty\}$, and either $\lim_{x\to x_0} g(x)=\infty$ or $\lim_{x\to x_0}f(x)=\lim_{x\to x_0} g(x)=0$. Then, $$\lim_{x\to x_0}\frac{f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2250517",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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SAT inequality problem I've been studying from Collegeboard SAT practice tests, and I've stumbled with a inequality problem, which I can't seem to understand even with SAT answer explanation.I would greatly appreciate it if anyone could help me.
$$ y ≤ 3x+1 $$
$$x-y > 1 $$
Which of the following ordered pairs (x, y) s... | $$y\leq3x+1$$ $$x-y>1\implies y<x-1$$
You can solve this graphically
The region which solves these inequalities is the region below the two lines. By plotting the 4 points, you'll see option D is the only one which lies in the region.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2250602",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
$2x^2 + 3x +4$is not divisible by $5$ I tried by $x^2 \equiv 0, 1, 4 \pmod 5$ but how can I deal with $3x$?
I feel this method does not work here.
| $$5|\;(2x^2+3x+4)\iff$$ $$ \iff5|\;3(2x^2+3x+4)=(6x^2+9x+12)\iff$$ $$\iff 5|\;((6x^2+9x+12)-(5x^2+5x+10))=$$ $$=(x^2+4x+2)=(x+2)^2-2.$$ But no square is $2$ more than a multiple of $5.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2250718",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 6,
"answer_id": 5
} |
Truncated normal random variable Find the cdf and quantile function for the truncated (at a) normal random variable given that $$\frac{\varphi(x) I_{x>a}}{1-\Phi(a)}$$ where $\varphi(x)$ is the density for standard normal and $\Phi(x)$ is the cdf for standard normal distribution. Express answers in terms of $\varphi(x)... | Since the cdf is $F_t (u)=\int_a^u\frac{\varphi(x) }{1-\Phi(a)} dx$, it can be expressed in terms of normal and hence standard normal (using the substitution $v=x-a $ to 'evaluate' the integral).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2250802",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
How many squares can be inscribed in a regular polygon? Say that a square $S$ is said to be inscribed in a regular polygon $P$ if all the four vertices of $S$ lie on the boundary of $P$. It is well-known that one can inscribe a square in a regular $n$-gon for $n\geq 5$.
I would like to know, up to rotational symmetry, ... | If $n$ is a multiple of $4$, then every couple of opposite points (with respect to the center) on the polygon can be taken as endpoints of a diagonal of an inscribed square, so in this case we have infinitely many solutions.
In the other cases, it is not difficult to prove that a solution is possible, where the inscrib... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2251079",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Snooker shot - does margin of error increase or decrease as the target angle increases? There is a perception (widely held) in snooker that a straight shot is more difficult than an angled shot. There are many forum discussion about this, and the reasons are usually accepted to be psychological.
But I was wondering, i... | Let the ball be radius $r$ and distance between balls $d$. Let $\theta$ be angle white is struck from line between centre of balls and $\phi$ direction red moves from line between red ball and white before struck. Then get $2r sin(\theta + \phi) = d sin(\theta)$. Call $a=\frac{d}{2r}$. This gives $\frac{d\phi}{d\theta}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2251189",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 0
} |
Where does the curve $x^y = y^x$ intersect itself? This problem is quite easily solved by using logarithms and derivative and forming the function $f(x,y) = x^y - y^x$, however there are assertions that this problem can be solved without using either of the two. I can not see how one would proceed to solve this problem... | Ok the function $f\colon \mathbb R^2 \to \mathbb R$ is continuous. And $f(1,2)=-1$ and $f(2,1)=1$ so it has to take the values between -1 and 1. Thus for some point we have that $f$ is zero.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2251318",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Maximize number of covered sets by choosing given number of elements I am not sure if it's a known probem:
There is a set of some elements. For the purpose of this explanation that can be a subset of natural numbers, let's say $\{1, 2, ..., 20\}$. Let's call it $SET$.
There are also given subsets of the $SET$. That sub... | Could someone explain the way that Mees de Vries answer meets my criteria. I think it is about searching smallest number of subsets covering SET, not about finding vertices that satisfy the maximum number of subsets. Correct me if I am wrong.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2251418",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Geometric series sum $\sum_{n=0}^\infty\frac{a}{(1+x)^n}$ Why does this hold
$$\sum_{n=0}^\infty\frac{a}{(1+x)^n}=\frac{a(x+1)}{x}$$
?
To me it looks like $=\frac{a}{x}$ from the formula.
That is:
$$r=(1+x)^{-n}\Rightarrow a(1-(1+x))^{-n}=a/n$$
| Okay, do if you put $n=0,1,2..$ in the forumula you'll get,
this series,
$a, \frac{a}{1+x}, \frac{a}{(1+x)^2}...$
with common ration $\frac{1}{1+x}$ now we can use the forumula for sum of $\infty$ GP Which is : $\frac{a}{1-r}$ here $r$ is common ratio;
so we get-$\frac{a}{1-\frac{1}{1-x}}$ which becomes $\frac{a(x+1)}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2251536",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Chess board probability problem. Three random squares are chosen from a regular chess board. Find the probability that they form the letter 'L'.
I cannot think about a general way to go about these type of questions. Need hints or solutions.
| There is a 1.92 % chance of making an "L" without space in between. If you mean that they can form an "L" when they are far from each other is a different story.
POSSIBLE Solution: 64/100=0.64 0.64x3=1.92,
And this was in percentage form. If you need the answer not in percentage and in something else i can`t help y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2251616",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 4
} |
existence of closest point on boundary of domain to a point outside domain Let $D \subset \mathbb{R}^n$ (or $\mathbb{C}^n$) be a domain with $C^2$ boundary. Why is there a neighborhood U of $\partial D$ such that for every $z \in U$, there is a unique point of $\partial D$ that is closest to $z$?
| This follows from the tubular neighbourhood theorem. Pick $U$ as a tubular neighbourhood of $\partial D$. Let $z \in U$, and let $p \in \partial D$ be a point which minimizes distance from $z$. It is easy to see that the segment joining $p$ and $z$ must be normal to $\partial D$, and thus the point $z$ corresponds to $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2251710",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
$p=(1-\zeta_{p})...(1-\zeta^{p-1}_{p})$ Let $\zeta_{p}$ be a primitive $p$th root of $1$. Then $t^{p}-1=(t-1)(t-\zeta_{p})(t-\zeta_{p}^{2})\ldots(t-\zeta_{p}^{p-1})$. Using this, I need to show that $p=(1-\zeta_{p})\ldots(1-\zeta^{p-1}_{p})$ but I do not know how.
| Divide by $t-1$, use that $\frac{t^p-1}{t-1}=1+t+\dotsb + t^{p-1}$ and then evaluate your equation at $t=1$.
For an alternative, you can evaluate the derivation of both sides at $t=1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2251825",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Is it possible to scale the eigenvector matrix up by multiplying or adding with a constant? I am using R to generate eigenvector matrix from laplacian matrix that represent a graph dataset. The issue that I have is that the values of eigenvector matrix are very much low, sometimes in the order of $10^{-20}$! My questio... | Yes, it is, since the space of all eigenvectors corresponding to an specific eigenvalue is a vector space, so if you multiply them by nonzero constant or even add them you will still get an eigenvector.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2251946",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
$\lim_{x \to 0} \frac{\log \left(\cosh\left(x^2-xc\right) \right)}{x^2}=\frac{c^2}{2}$ without L'Hospital's rule How to show that without using L'Hospital's rule
\begin{align}
\lim_{x \to 0} \frac{\log \left(\cosh\left(x^2-xc\right) \right)}{x^2}=\frac{c^2}{2}
\end{align}
I was able to show the upper bound by using th... | Denote $a=x^2-cx$ for simplicity. Then
\begin{align}
\frac{\ln(\cosh a )}{x^2}&=\frac{\ln(\cosh^2a)}{2x^2}=\color{blue}{\frac{\ln(1+\sinh^2a)}{2x^2}}=\frac12\cdot\frac{\ln(1+\sinh^2a)}{\sinh^2a}\cdot\left(\frac{\sinh a}{x}\right)^2=\\
&=\frac12\cdot\frac{\ln(1+\sinh^2a)}{\sinh^2a}\cdot\left(\frac12\cdot\left[\frac{e^a-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2252022",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 1
} |
Let $f,g: V \rightarrow W$ be linear operators. Prove that $r(f + g) \leq r(f) + f(g)$ Let $f,g: V \rightarrow W$ be linear operators.
Prove that $r(f + g) \leq r(f) + f(g)$
Note: r = rank
My idea was to use matrix representation and to prove that the rank of matrix C ($C= F+G$) can't be bigger than the sum of $r(F) +... | Hint: $\operatorname{im}(f+g)\subseteq \operatorname{im}f+\operatorname{im}g$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2252161",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Definition of a continuous function (through open sets) Why the definition of a continuous function defined in terms of inverse function?
$f$ - continuous function, if for every open $V$, $f^{-1}(V)$ is open.
| Because it is easier to use than the regular caculus like definition,
f is continuous iff for all x and open V nhood f(x), there be an open U nhood of x with f(U) subset V.
The two defintions can be proven equivalent.
The later for real to real functions can be be shown equivalent to the usual epsilon delta definiti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2252261",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
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