Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Estimate total distance using Riemann sum based on data table We did not spend a lot of time in class on Riemann sum so I confused with this question.
Speedometer readings for a motorcycle at $12$-second intervals are given in the table below.
$$
\begin{array}{c|c|c|c|c}
t sec & 0 & 12 & 24 & 36 & 48 & 60 \\
\hline
v(t... | The definition of a Riemann sum is as following:
Let $f$ be a function, $\Pi=\{x_0,\dots,x_n\}$ be a partition and $S=\{c_1,\dots,c_n\}$ a set of values such that $c_i\in[x_{i-1},x_i]$. The Riemann sum is
$$R(\Pi,S)=\sum_{i=1}^nf(c_i)(x_i-x_{i-1}).$$
You have a discrete function $f=v$ and you have a partition in first ... | {
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"url": "https://math.stackexchange.com/questions/1066355",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Calculate: $ \lim_{x \to 0 } = x \cdot \sin(\frac{1}{x}) $ Evaluate the limit:
$$ \lim_{x \to 0 } = x \cdot \sin\left(\frac{1}{x}\right) $$
So far I did:
$$
\lim_{x \to 0 } = x\frac{\sin\left(\frac{1}{x}\right)}{\frac{1}{x}\cdot x}
$$
$$
\lim_{x \to 0 } = 1 \cdot \frac{x}{x}
$$
$$
\lim_{x \to 0 } = 1
$$
Now of course ... | Your proof is incorrect, cause you used incorrect transform, but it has already been stated. I'll describe way to solve it.
$$\lim_{x \to 0}\frac{\sin(\frac{1}{x})}{\frac{1}{x}} \neq 1$$
Hint: Solution is well-known trick. Note $(\forall x \in \mathbb{R})\left(\sin(x) \in[-1;1]\right)$ (obvious) and use squeeze theorem... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1066434",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Are finitely presentable modules closed under extensions?
If $0 \to A \to B \to C \to 0$ is an exact sequence of modules, and $A$ and $C$ are finitely presentable, then is $B$ finitely presentable?
The answer is "yes" if we replace modules with groups, as shown here.
The answer is also "yes" if we replace "finitely ... | You can proceed as usual by starting with $F\stackrel{f}\to A\to 0$ and $H\stackrel{h}\to C\to 0$, where $F$ and $H$ are free of finite rank. Then show that there is an exact sequence $G=F\oplus H\stackrel{g}\to B\to 0$. Now consider $F'=\ker f$ and so on. You have a short exact sequence $0\to F'\to G'\to H'\to 0$. Now... | {
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"url": "https://math.stackexchange.com/questions/1066604",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
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Question about sines of angles in an acute triangle
Let $\triangle ABC$ be a triangle such that each angle is less than $ 90^\circ $.
I want to prove that $\sin A + \sin B + \sin C > 2$.
Here is what I have done:
Since $A+B+C=180^{\circ}$ and $0 < A,B,C < 90^\circ$, at least two of $A,B,C$ are in the range 45 < x <... | i am able to simplify $$
\sin A + \sin B + \sin (A + B) = \sin A + \sin B + \sin A \cos B + \sin B \cos A \\ = (1+\cos B)\sin A + (1 + \cos A)\sin B
\\ = 4\cos^2 B/2\sin A/2 \cos A/2 + 4\cos^2 A/2 \sin B/2 \cos B/2
\\ = 4\cos B/2 \cos A/2(\sin A/2 \cos B/2 + \sin B/2 \cos A/2)
\\ = 4\cos B/2 \cos A/2\sin (A/2 + B/2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1066712",
"timestamp": "2023-03-29T00:00:00",
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The Centralizer $C_H(x)$ where $x \in G$ and $H \leq G$. Let $G$ be a group and $H$ be a subgroup of $G$. Let $x \in G$. Then
$C_H(x)=H$ if and only if $x \in Z(H)$?
It is obvious that if $x \in Z(H)$ then $C_H(x) = H$.
But I could not prove or provide a counter-example to the other statement.
Remark: $C_H(x) = \{h\in... | Consider the example of $G=\mathbb{Z}_2\times \mathbb{Z}_2$, $H=\mathbb{Z}_2\times \{0\}$. Then $(0,1)$ satisfies $C_H((0,1))=H$, however $(0,1)\notin Z(H)$ because $(0,1)\notin H$.
| {
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Preserve self-adjoint properties I was thinking about this problem recently:
Let $T$ be a self-adjoint operator on $L^2((-1,1),d x)$. Now you define an operator $G$ by $G(f) := T(\frac{f}{(1-x^2)})$ with $\operatorname{dom}(G):=\{f \in L^2(-1,1); \frac{f}{(1-x^2)} \in \operatorname{dom}(T)\}$. Is this operator also se... | Essentially, you need to check that $\forall f,g\in dom(G)\cap dom(T)$ you have
$$(f,T[g]/\phi) = (f,T[g/\phi]),$$
where $(\cdot,\cdot)$ is a scalar product in $L^2$ and $\phi$ is the function $\frac{1}{1-x^2}$.
I don't quite see how it could be possible for generic $T$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1066918",
"timestamp": "2023-03-29T00:00:00",
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Proving the limit at $\infty$ of the derivative $f'$ is $0$ if it and the limit of the function $f$ exist. Suppose that $f$ is differentiable for all $x$, and that $\lim_{x\to \infty} f(x)$ exists.
Prove that if $\lim_{x\to \infty} f′(x)$ exists, then $\lim_{x\to \infty} f′(x) = 0$, and also, give an example where $\li... | If $\lim_{x\rightarrow\infty}f'(x)=c$ were some positive number, that would imply, for some $0<k<c$ and all large enough $x$ that $f'(x)>k$. Think about what this means intuitively and why this is inconsistent with $f$ converging.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1067005",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Partial Converse of Holder's Theorem Holder's Theorem is the following: Let $E\subset \mathbb{R}$ be a measurable set. Suppose $p\ge 1$ and let $q$ be the Holder conjugate of $p$ - that is, $q=\frac{p}{p-1}.$ If $f\in L^p(E)$ and $g\in L^q(E),$ then $$\int_E \vert fg\vert\le \Vert f\Vert _p\cdot\Vert g \Vert_q$$
I am t... | An alternative proof.
Let $T_g(f):=\int_E f g\,,\, \forall f\in L^p.$
It's clear that $T_g$ is linear.
By the condition $\int_E \vert fg\vert\le M\Vert f\Vert_p$, we see $T_g$ is bounded.
So, $T_g\in (L^p)^*.$
By the Riesz Representation Theorem for the Dual of $L^p$(c.f. p. 160, Real Analysis, 4th Edition, Royden), we... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Explanation for the definition of monomials as products of products I'm attempting to learn abstract algebra, so I've been reading these notes by John Perry. Monomials are defined (p. 23) as
$$
\mathbb{M} = \{x^a : a \in \mathbb{N}\} \hspace{10mm} \text{or} \hspace{10mm} \mathbb{M}_n = \left\{\prod_{i=1}^m{\left( x_1^{... | In the commutative case, $x_1x_2x_1$ is equal to $x_1^2x_2$. However, in the non-commutative case, they need not be equal. Suppose now that we're in a situation where $x_1x_2x_1 \neq x_1^2x_2$. If the set of monomials had not been defined as products of products, then $x_1x_2x_1$ would not be considered a monomial as i... | {
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Calculating the mean and variance of a distribution
*
*Suppose $$P(x) = \frac{1}{\sqrt{2\pi\cdot 36}}e^{-\frac{1}{2}\cdot (\frac{x-2}{6})^2}$$
What is the mean of $X$? What is the standard deviation of $X$?
*Suppose $X$ has mean $4$ and variance $4$. Let $Y = 2X+7$.
What is the mean of $Y$? What is the standard... | 1) Since: $X \sim N(\mu, \sigma^2)$ you have that:
$$\mu=2 \qquad \sigma^2=6^2$$
2) $$E[Y]=E[2X+7]=E[2X]+E[7]=2E[X]+E[7]$$
Substituting $E[X]$ with $\mu=4$, we get: $$E[Y]=2 \cdot 4+7=15$$
For what it concerns the variance: $$Var[Y]=\cdots=Var[2X]+Var[7]=2^2\cdot Var[X]+Var[7]$$
Substituting $Var[X]=\sigma^2=4$, and si... | {
"language": "en",
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Is this a legitimate proof? If not, how to prove? Question: Determine all natural numbers $n$ such that: $7 \mid \left(3^n - 2\right)
\implies3^{n}\equiv 2\pmod{7}$
Multiply both sides by 7
$7 \cdot 3^{n}\equiv 7\cdot2\pmod{7}$
Divide both sides by seven, since $\gcd(7,7) = 7$, we have to divide modulus by $7$
$\implie... | Noting that $n=1$ does not work, let $n \ge 2$. Then as $3^2 \equiv 2 \pmod 7$, we have the equivalent statement
$$2\cdot 3^{n-2} \equiv 2 \pmod 7 \iff3^{n-2}\equiv 1 \pmod 7$$
Now that has solutions $n = 6k+2$ as $3^6$ is the smallest positive power of $3$ that is $\equiv 1 \pmod 7$, so the solution is for natural nu... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Compute almost sure limit of martingale?
Let $Y$, $Y_1$, $Y_2$, $\dots$, be nonnegative i.i.d random variables with mean $1$. Let
$$X_n = \prod_{1\le m \le n}Y_m$$
If $P(Y = 1) < 1$, prove that $\lim\limits_{n\to\infty}X_n = 0$ almost surely.
I feel like this question has something to do with the idea that $(X_... | $\dfrac{\log X_n}{n} = \dfrac{\sum_{m \le n} \log Y_m}{n} \to E\log Y_1$ almost surely by the strong law of large number.
And by Jensen's inequality, $E\log Y_1 \leq \log EY_1 =0$ since $EY_1 = 1$.
Since $P(Y = 1) < 1$, $E\log Y_1 < \log EY_1 =0$.
So we get that $\dfrac{\log X_n}{n}$ converges to a strictly negative nu... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Rules for whether an $n$ degree polynomial is an $n$ degree power Given an $n$ degree equation in 2 variables ($n$ is a natural number)
$$a_0x^n+a_1x^{n-1}+a_2x^{n-2}+\cdots+a_{n-1}x+a_n=y^n$$
If all values of $a$ are given rational numbers, are there any known minimum or sufficient conditions for $x$ and $y$ to have:
... | This addresses user45195's question and is too long for a comment.
When I said too broad, it was because the question originally didn't limit the field of $x$. A familiar field is the complex numbers $\mathbb{C}=a+bi$, of which a special case are the reals $\mathbb{R}$, and even more limited, the rationals $\mathbb{Q}... | {
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Is it possible to find the $n$th digit of $\pi$ (in base $10$)? Is it possible that there exists some function $f:\mathbb N_1\to \{0,1,2,3,4,5,6,7,8,9\}$, where $$f(1)=\color{red}1, f(2)=\color{red}4, f(3)=\color{red}1, f(4)=\color{red}5, f(5)=\color{red}9, f(6)=\color{red}2, \ldots \quad ?$$
I know that such a functio... | What exactly means explicit? For a given $n$, there is certainly an algorithm that computes the $n$'th digit of $\pi$.
But the answer is no if you ask about some other real numbers. There are non-computable numbers that encode the halting problem, for example. There is no chance to have an algorithm computing the digi... | {
"language": "en",
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How many positive integers of n digits chosen from the set {2,3,7,9} are divisible by 3? I'm preparing myself for math competitions. And I am trying to solve this problem from the Romanian Mathematical Regional Contest “Traian Lalescu’', $2003$:
Problem $\mathbf{7}$: How many positive integers of $n$ digits chosen fro... | Here is an alternative approach. Let $x_n,y_n$, and $z_n$ be as in the argument given in the question; clearly $x_1=2$, and $y_1=z_1=1$. For $n\ge 1$ let $X_n,Y_n$, and $Z_n$ be the sets of $n$-digit numbers using only the digits $2,3,7$, and $9$ and congruent modulo $3$ to $0,1$, and $2$, respectively (so that $x_n=|X... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Let $A$ be a non-zero linear transformation on a real vector space $V$ of dimension $n$.
Let $A$ be a non-zero linear transformation on a real vector space $V$ of dimension $n$. Let the subspace $V_0 \subset V$ be the image of $V$ under $A$. Let $k = \dim (V_0) \lt n$ and suppose that for some $\lambda \in \mathbb{R}$... | If $\lambda\ne0$ then the polynomial with simple roots $x^2-\lambda x=x(x-\lambda)$ annihilates $A$ and clearly $A\ne \lambda I_n$ and $A\ne0$ so $0$ and $\lambda$ are eigenvalues of $A$ and the multiplicity of $\lambda$ is $k$. If $\lambda=0$ then $A$ is nilpotent and $A$ in it's Jordan canonical form has a Jordan blo... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Geometric distribution example (making kids until couple has a boy and a girl), need explanation So the condition is following: a man and a woman want to have kids : girl and a boy. They continue to make kids until they get both genders. What is the expected number of kids?
As I remember, the solution was following:
$$... | The given expression $$E[X]=1+\frac1p=3$$ can be explained as follows
*
*The term "$1$" stands for the first kid (if you want to have two kids you have to start by having the first kid). This kid has certainly a gender (is boy or girl) so you have the one gender after the first try! (great).
*Now, you have to keep ... | {
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"url": "https://math.stackexchange.com/questions/1067989",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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interpolation properties of analytic paths Assume we are given $n$ points in $\mathbb{C}^k$ can we find an analytic path
$\phi:[0,1]\to \mathbb{C}^k$ passing through these $n$ points?
| As Harald Hanche-Olsen pointed out, there is an interpolating polynomial of degree at most $n-1$ that does the job. Namely, let $p_1,\dots,p_n$ be the given points, pick any numbers $0\le t_1<t_2<\dots<t_n\le 1$, and define
$$
\phi(t) = \sum_{j=1}^n p_j \frac{\prod_{l\ne j}(t-t_l)}{\prod_{l\ne j}(t_j-t_l)}
$$
This is ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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lagrange interpolation, polynomial of degree $2n-1$ Let $a_1, \dots, a_n$ and $b_1, \dots, b_n$ be real numbers. How would I go about showing the following?
*
*If $x_1, \dots, x_n$ are distinct numbers, there is a polynomial function $f$ of degree $2n - 1$, such that $f(x_j) = f'(x_j) = 0$ for $j \neq i$, and $f(x_i... | 1.
Clearly $f$ will have to be of the form$$f(x) = \prod_{\substack{j=1 \\ j\neq i}}^n (x- x_j)^2(ax+b)$$$($because each $x_j$, $j \neq i$ is a double root$)$. It therefore suffices to show that $a$ and $b$ can be picked so that $f(x_i) = a_i$ and $f'(x_i) = b_i$. If we write $f$ in the form $f(x) = g(x)(ax + b)$, then... | {
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"timestamp": "2023-03-29T00:00:00",
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Calculate the distance between intersection points of tangents to a parabola
*
*Question
Tangent lines $T_1$ and $T_2$ are drawn at two points $P_1$ and $P_2$ on the parabola $y=x^2$ and they intersect at a point $P$. Another tangent line $T$ is drawn at a point between $P_1$ and $P_2$; it intersects $T_1$ at $Q_1$ a... | Suppose that the third tangent is drawn at a point $A$ with coordinates $A(a, a^2)$. Then its tangent intersects $T_{i}$ at
$$
Q_{i}\left(\frac{P_{ix} + a}{2}, P_{ix}a \right)
$$
using your equation $(3)$. In other words,
$$
2(Q_{1x} - Q_{2x})=P_{1x}-P_{2x}
$$
Therefore, in the argument of the square root in the numer... | {
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"timestamp": "2023-03-29T00:00:00",
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Is $[\bar{\mathbb Q}:\bar{\mathbb Q}\cap\mathbb R]=2$?
Is $[\bar{\mathbb Q}:\bar{\mathbb Q}\cap\mathbb R]=2$ ?
I think it is true that $\bar{\mathbb Q}\cap\mathbb C=\bar{\mathbb Q}$, because I've heard that the closure of the reals is $\mathbb C$. And $\bar{\mathbb Q}$ is a subfield of $\mathbb C$, so using this fact... | Yes, $[\overline{\mathbb Q}:\overline{\mathbb Q}\cap \mathbb R]=2$, since you attain the former from the latter by adjoining $i$.
The index is always either a positive integer or infinite, since it represents the cardinality of a basis of a vector space.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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How to solve a linear system in matrix form using Laplace transform? How to solve this linear system using Laplace transform?
$$\mathbf X'(t)=\left[\begin{array}{r,r,r}-3&0&2\\1&-1&0\\-2&-1&0\end{array}\right]\mathbf X(t); ~~~~~~~~\mathbf X(0)=\left[\begin{array}{r}4\\-1\\2\end{array}\right]$$
I am struggling with this... | We are given:
$$X'(t) = \begin{bmatrix} -3 & 0 & 2 \\ 1 & -1 & 0\\ -2 & -1 & 0\end{bmatrix} \begin{bmatrix} x(t) \\ y(t)\\ z(t)\end{bmatrix}, ~~ X(0) = \begin{bmatrix} 4 \\ -1\\ 2\end{bmatrix}$$
We can write this as:
$$\tag 1 \begin{align} x' &= -3x + 2z \\ y' &= x-y \\ z' &= -2x - y \end{align}$$
Taking the Laplace tr... | {
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Order of $f(n) = 4n + 6n^3 - 8n^5$ If a function $$f(n) = 4n + 6n^3 - 8n^5$$ then the order of $f$ is:
The answer I have is $\log(n)$, but I'm not sure if it's right.
| The order of a polynomial is usually its largest power. In this case, it would be 5.
If instead you are trying to find $g$ s.t.
$$
f\in O\left(g\right)\text{ as }n\rightarrow\infty,
$$
(big O notation) then $g$ can be $n^{5}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1068572",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Evaluating $\int_{0}^{\pi/2}\frac{x\sin x\cos x\;dx}{(a^{2}\cos^{2}x+b^{2}\sin^{2}x)^{2}}$ How to evaluate the following integral
$$\int_{0}^{\pi/2}\frac{x\sin x\cos x}{(a^{2}\cos^{2}x+b^{2}\sin^{2}x)^{2}}dx$$
For integrating I took $\cos^{2}x$ outside and applied integration by parts.
Given answer is $\dfrac{\pi}{4ab^... | let $x=\dfrac{t}{2}$, we have
$$I=\dfrac{1}{2}\int_{0}^{\pi}\dfrac{\dfrac{t}{2}\sin{\dfrac{t}{2}}\cos{\dfrac{t}{2}}}{\left(a^2\sin^2{\dfrac{t}{2}}+b^2\cos^2{\dfrac{t}{2}}\right)^2}dt=\dfrac{1}{2}\int_{0}^{\pi}\dfrac{t\sin{t}}{[(a^2+b^2)+(a^2-b^2)\cos{t}]^2}dt$$
So
\begin{align*}I&=-\dfrac{1}{2(a^2-b^2)}\int_{0}^{\pi}t\... | {
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NP Solvable in Polynomial Time I just took an exam and am a little curious about this question (it may not be verbatim, but the idea is clear):
TRUE/FALSE: If an NP complete problem can be solved in polynomial time, then P = NP.
My thought was FALSE. A single NP-complete problem being solved in polynomial time doesn't... | You're incorrect. If some NP-complete problem $A$ can be solved in polynomial time, then given any other NP problem $B$ we can solve it in polynomial time by first reducing $B$ to $A$ in polynomial time and then running the polynomial algorithm for $A$. A nearby statement is false, namely, that if a problem in NP can b... | {
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Power set of Set differences
Assume that $\mathcal P(A-B)= \mathcal P(A)$. Prove that $A\cap B = \varnothing$.
What I did:
I tried proving this directly and I got stuck.
Let $X$ represent a nonempty set, and let $X\in\mathcal P(A-B)$.
By definition of power set and set difference:
$X\subseteq A$ and $X\nsubseteq B$.
... | We will prove the contrapositive:
If $A \cap B \neq \emptyset$, then $\mathcal P(A - B) \neq \mathcal P(A)$.
Suppose that there is some element $x \in A \cap B$ so that $x \in A$ and $x \in B$. Then $x \notin A - B$ (otherwise, if $x \in A - B$, then $x \notin B$, contradicting the fact that $x \in B$). This implies ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1068811",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Showing that $f(x)=x^2$ for $x \in \mathbb{Q}$ and $f(x)=0$ for $x \not\in \mathbb{Q}$ is differentiable in $x=0$ I am supposed to show that $f(x) = x^2$ for $x$ in the rationals and $f(x) = 0$ for $x$ in the irrationals is differentiable at $x = 0$ and I am supposed to find the derivative of $f(x)$ at $x = 0$.
Is my p... | You're pretty close, but you're missing the case when $x$ is irrational.
To find the derivative we have to evaluate $$f'(0)=\lim\limits_{x\rightarrow 0}\frac{f(x)-f(0)}{x-0}=\lim\limits_{x\rightarrow 0}\frac{f(x)}{x}$$ since $f(0)=0^2=0$. One way to evaluate this is to let $(x_n)\rightarrow 0$ be an arbitrary sequence ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Metric Spaces: closure of a set is the set of all limits of sequences in that set I am studying metric spaces and got confused about many different ways of defining the closure.
Let $S$ be a subset of $M.$ Then, the closure of $S$ is $ \{x \in M : \forall \epsilon>0, \ \ B(x,\epsilon) \cap S \neq \emptyset \}$
Also, t... | The property of a point $x$ that
for all $\varepsilon>0$, the intersection $B(x,\varepsilon)\cap S$ is non-empty
is equivalent to
every neighborhood $U$ of $x$ intersects $S$
since $B(x,\varepsilon)$ is a neighborhood, and every neighborhood $U$ contains a ball $B(x,\varepsilon)$ for some $\varepsilon>0$. These cha... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A question about matrices such that the elements in each row add up to $1$.
Let $A$ be an invertible $10\times 10$ matrix with real entries such that the sum of each row is $1$. Then is the sum of the entries of each row of the inverse of $A$ also $1$?
I created some examples, and found the proposition to be true. I ... | Any square matrix $\;n\times n\;$ has rows sum equal to $\;1\;$ iff $\;u:=(1,1,\ldots,1)^t\;$ is an eigenvector with eigenvalue $\;1\;$, but then
$$Au=u\implies A^{-1}\left(Au\right)=A^{-1}u\iff u= A^{-1}u$$
so the answer is yes.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Intersection of two circles. Let $C_1$ and $C_2$ be the circles: $\rho=a\sin\theta, \rho=a(\cos\theta + \sin\theta)$ respectively. The graphs of these two circles are
From the graphs, we see that the intersection points are $(0,0)$, $(\pi/2, a)$. But when we solve the system of equations: $\rho=a\sin\theta, \rho=a(\co... | Note that $\left(-\frac{\pi}{2},-a\right)$ is another representation of the point $\left(\frac{\pi}{2},a\right)$ (to see this, draw the radius $-a$ at an angle of $-\frac{\pi}{2}$). So the two points you get are actually the same point written differently.
As for $(0,0)$, note that $a\sin\theta$ passes through $(0,0)$ ... | {
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"timestamp": "2023-03-29T00:00:00",
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Find an $\epsilon$ such that the $\epsilon$ neighborhood of $\frac{1}{3}$ contains $\frac{1}{4}$ and $\frac{1}{2}$ but not $\frac{17}{30}$ I am self studying analysis and wrote a proof that is not confirmed by the text I am using to guide my study. I am hoping someone might help me comfirm/fix/improve this.
The prob... |
$ϵ<\frac{1}{15}$ is a satisfactory solution.
It's best not to use same letter to mean two different things: the $\epsilon$ that's requested in the problem is somehow also $\frac16+\epsilon$ at the end of proof. You could write $\epsilon = \frac16+\delta$. So, your final answer is
$$\frac16<\epsilon<\frac{1}{6}+\frac... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Closed form of $\int_0^\infty \ln \left( \frac{x^2+2kx\cos b+k^2}{x^2+2kx\cos a+k^2}\right) \;\frac{\mathrm dx}{x}$ Today I discussed the following integral in the chat room
$$\int_0^\infty \ln \left( \frac{x^2+2kx\cos b+k^2}{x^2+2kx\cos a+k^2}\right) \;\frac{\mathrm dx}{x}$$
where $0\leq a, b\leq \pi$ and $k>0$.
... | $\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,}
\newcommand{\dd}{{\rm d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{... | {
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Erwin Kreyszig's Introductory Functional Analysis With Applications, Section 2.7, Problem 6
Suppose that $X$ and $Y$ are two normed spaces over the same field ($\mathbb{R}$ or $\mathbb{C}$). Show that the range of a bounded linear operator $T \colon X \to Y$ need not be closed in $Y$.
Kreyszig gives the following hi... | Let $e_n$ be the element of $\ell_\infty$ whose $m$'th coordinate is $1$ if $m=n$ and $0$ otherwise. The closed linear span of $\{e_n\mid n\in \Bbb N\}$ in $\ell_\infty$ is the space $c_0$ of sequences that tend to $0$.
For your operator, we have $T(je_j)=e_j$; so the range of $T$ contains each $e_j$. Since the range ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proof $|\sin(x) - x| \le \frac{1}{3.2}|x|^3$ So, by Taylor polynomial centered at $0$ we have:
$$\sin(x) = x-\frac{x^3}{3!}+\sin^4(x_o)\frac{x^4}{4!}$$
Where $\sin^4(x_0) = \sin(x_o)$ is the fourth derivative of sine in a point $x_0\in [0,x]$.
Then we have:
$$\sin(x)-x = -\frac{x^3}{3!}+\sin(x_0)\frac{x^4}{4!}$$
I thou... | $$\sin x - x = \int_{0}^{x}(\cos t-1)\, dt$$
hence, for any $x>0$:
$$|\sin x - x| = \int_{0}^{x}2\sin^2\frac{t}{2}\,dt\leq \int_{0}^{x}\frac{t^2}{2}\,dt=\frac{x^3}{6}$$
since $\sin\frac{t}{2}\leq\frac{t}{2}.$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is indefinite integration non-linear? Let us consider this small problem:
$$
\int0\;dx = 0\cdot\int1\;dx = 0\cdot(x+c) = 0
\tag1
$$
$$
\frac{dc}{dx} = 0 \qquad\iff\qquad
\int 0\;dx = c, \qquad\forall c\in\mathbb{R}
\tag2
$$
These are two conflicting results. Based on this other question, Sam Dehority's answer seems to ... | My "answer" should be a comment under the two posted answers, but I don't have enough reputation to post comments. There is a (very popular) mistake in both answers. Indefinite integral operator does NOT give a class of functions equal up to constant translation. Let's look at an example. Someone might evaluate integra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1069664",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "27",
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Taylor Expansion of $x\sqrt{x}$ at x=9 How can I go about solving the Taylor expansion of $x\sqrt{x}$ at x=9?
I solved the derivative down to the 5th derivative and then tried subbing in the 9 value for a using this equation
$$\sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n.$$
Can someone walk me through the process (i... | Lets write down the first couple of derivatives first:
$f(x)=x\cdot \sqrt x$
$f'(x)=\frac{3\cdot\sqrt x }{2}$
$f''(x)=\frac{3}{4\cdot \sqrt x}$
You mentioned the taylor expansion in your opening post:
$\sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n.$
So your first three terms will be:
$\frac{f^{o}(9)}{0!}(x-9)^0=27$
$\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Limits using Maclaurins expansion for $\lim_{x\rightarrow 0}\frac{e^{x^2}-\ln(1+x^2)-1}{\cos2x+2x\sin x-1}$ $$\lim_{x\rightarrow 0}\frac{e^{x^2}-\ln(1+x^2)-1}{\cos2x+2x\sin x-1}$$
Using Maclaurin's expansion for the numerator gives:
$$\left(1+x^2\cdots\right)-\left(x^2-\frac{x^4}{2}\cdots\right)-1$$
And the denominator... | In a case where $x \to 0$ as here, if the two expansions of top and bottom happen to start with the same degree, the limit is the ratio of the coefficients. Here, the top expansion starts with $x^4$ while the bottom starts with $x^4/3$ making the limit $3.$
That is, we have
$$\frac{x^4+\cdots}{x^4/3+\cdots},$$
and whe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1069824",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Quadratic formula in double inequalities I have the double inequality:
$-x^2 + x(2n+1) - 2n \leq u < -x^2 + x(2n-1)$
and I am trying to get it into the form
$x \leq \text{ anything } < x+1$
Or at least solve for x as the smallest term. I know I need to use the quadratic formula but I don't understand how I can solve... | If you write
$$f_1(x)=x^2-(2n+1)x+2n+u\ ,\quad f_2(x)=x^2-(2n-1)x+u$$
then you want to solve
$$f_1(x)\ge0\ ,\quad f_2(x)<0\ .$$
The important thing to notice is that
$$f_2(x)=f_1(x+1)\ .$$
Each quadratic has discriminant
$$\Delta=(2n-1)^2-4u\ .$$
Firstly, if $\Delta\le0$ then $f_2(x)$ cannot be negative, and so there i... | {
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Branch cut and principal value I do not understand the principal value and it is relation to branch cut.
Please tell me about principal value with some examples, then explain the branch cut concept.
For instance, what is the $\text{Arg} (-1-i)$ , tell me your thinking steps.
Thanks
| In Complex Analysis, we usually have defined $\arg(z)$ and $\text{Arg}(z)$ where the later generally denotes the principal argument. Most books that I have dealt with define the principal argument to lie in $(-\pi, \pi)$ but it is not unheard of to see it defined between $(0, 2\pi)$.
If we consider $z = -1 - i$, then w... | {
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Find the remainder if $19^{55}$ is divided by 13. The question, as stated in the title, is
Find the remainder if $19^{55}$ is divided by 13.
Here is my approach for solving this problem.
I know that $19\equiv6$ (mod 13), so $19^{55}\equiv 6^{55}$ (mod 13). Then I can see that $6^{12}\equiv 1$ (Fermat's Little Theorem... | $6^7\equiv 6(6^2)^3\pmod{13}\equiv 6(-3)^3\pmod{13}\equiv -6.27\pmod{13}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1070091",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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A power series that converges for $|x| \leq 1$ and diverges otherwise. I need to find a power series $\sum a_n z^n$ that converges for $|x| \leq 1$ and diverges otherwise.
I think I have one I just want to be sure.
So, the series:
$\sum \frac{z^n}{n^2}$
has radius of convergence of 1. So it converges when $|z| <1$ ... | You don't need the comparison test. If $|z|=1$ then $\sum z^2/n^2$ converges absolutely so it converges.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1070154",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Linear Algebra: Polynomials Basis Consider the polynomials $$p_1(x) = 1 - x^2,\;p_2(x) = x(1-x),\;p_3(x) = x(1+x)$$
Show that $\{p_1(x),\,p_2(x),\,p_3(x)\}$ is a basis for $\Bbb P^2$.
My question is how do you even go about proving that these polynomials are even independent? Are there certain rules I should know?
| No particular rule for polynomials: they are elements of a vector space of dimension $3$.
Since $\{1;x;x^2\}$ is obviously a basis for your space, you can simply show that the matrix
$$
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 1 \\
-1 & -1 & 1
\end{bmatrix}
$$
has rank $3$, which is done by a simple elimination.
Why is thi... | {
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Series sum $\sum 1/(n^2+(n+1)^2)$ In an exercise, I caculate the Fourier expansion of $e^x$ over $[0,\pi]$ is
$$e^x\sim \frac{e^\pi-1}{\pi}+\frac{2(e^\pi-1)}{\pi}\sum_{n=1}^\infty \frac{\cos 2nx}{4n^2+1}+\frac{4(1-e^\pi)}{\pi}\sum_{n=1}^\infty \frac{n\sin 2nx}{4n^2+1}.$$
From this, it is easy to deduce
$$\sum_{n=1}^\i... | We can approach such kind of series by considering logarithmic derivatives of Weierstrass products. For instance, from:
$$\cosh z = \prod_{n=1}^{+\infty}\left(1+\frac{4z^2}{(2n-1)^2\pi^2}\right)\tag{1}$$
we get:
$$\frac{\pi}{2}\tanh\frac{\pi z}{2} = \sum_{n=1}^{+\infty}\frac{2z}{z^2+(2n-1)^2}\tag{2},$$
so, evaluating i... | {
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"timestamp": "2023-03-29T00:00:00",
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Discrete analogue of Green's theorem Following formula concerning finite differences is in a way a discrete analogue of the fundamental theorem of calculus:
$$\sum_{n=a}^b \Delta f(n) = f(b+1) - f(a) $$
We can think about the Green's theorem as a two-dimensional generalization of fundamental theorem of calculus, so I'm... | It's entirely about summation: in a partition of a plane region $R$, we have the following for any function $G$ defined on the edges of the partition:
$$\sum_{\partial R} G=\sum_{R} \Delta G$$
($\Delta G$ is defined on the faces of the partition).
| {
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"source": "stackexchange",
"question_score": "8",
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Calculate $\lim_{x \to 0} (e^x-1)/x$ without using L'Hôpital's rule Any ideas on how to calculate the limit of $(e^x -1)/{x}$ as $x$ goes to zero without applying L'Hôpital's rule?
| I don't know if this is really "without" Hôpitals rule for you but if you are allowed to use
$$
\exp(x)=\sum_{k=0}^\infty \frac{x^k}{k!}
$$
the limit is straightforward since this sum converges locally uniformly.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1070524",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Why is this change of basis useful? In my textbook there is a theorem which states
Let $A$ be a real $2\times 2$ matrix with complex eigenvalues $\lambda =a\pm bi$ (where $(b\ne 0)$. If $\mathbf x$ is an eigenvector of $A$ corresponding to $\lambda=a-bi$, then the matrix $P=\begin{bmatrix} \operatorname{Re}(\mathbf x... | You can write $$C=\alpha\left[\begin{array}{cc}\cos\theta&-\sin\theta\\ \sin\theta &\cos\theta\end{array}\right]$$
$C^n$ involves $n\theta$ so it is easy to calculate, and hence powers of $A$ are easy to calculate.
On the other hand, I don't know how useful it is because I hadn't heard of it before.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Calculating canonical divisor in product of projective spaces. Let $X$ be an intersection of two divisors of bidegree $(a,b)$ and $(c,d)$ in $\mathbb{P^2}\times \mathbb{P^2}$. Then how can I find the canonical divisor $K_X$?
I'm asking because I have no experience in working with bidegrees.
| As usual when you have an intersection like this (which I assume has the right dimension and so on), you can use the adjunction formula.
Adjunction says that we have an exact sequence of bundles on $X$
$$ 0 \rightarrow N_X^\vee \longrightarrow \Omega_{\mathbf P^2 \times \mathbf P^2 \mid X} \rightarrow \Omega_X \longrig... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Triangulation of hypercubes into simplices A square can be divided into two triangles. A 3-dimensional cube can be divided into 6 tetrahedrons. Into what number of simplices an n-dimensional hypercube can be divided? (For example, a 4-hypercube, or a 5-hypercube.)
| As mentioned before there are (unsurprisingly) many ways to triangulate the cube. However one easy (and sometimes useful) way to do this is: Let $\pi$ be a permutation of $\{1,2,...,d\}$. Then define
$S_\pi = \{x \in \mathbb{R}^d: 0 \leq x_{\pi(1)} \leq x_{\pi(2)}\leq ... \leq x_{\pi(d)} \leq 1 \}$. Clearly, $S_\pi$ is... | {
"language": "en",
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$A$ has full rank iff $A^H A$ is invertible Let $A \in \mathbb{K}^{m,n}$ be a matrix. How to show that $\text{rank}(A) = n$ if and only if the matrix $A^HA$ is invertible?
| Unsure of your notation/assumptions, but here's a hint:
*
*For real matrices, $$\text{rank}(A^*A)=\text{rank}(AA^*)=\text{rank}(A)=\text{rank}(A^*)$$
*For complex matrices, $$\text{rank}(A^*A)=\text{rank}(A)=\text{rank}(A^*)$$
Mouse over for more after you've pondered it a bit:
$\text{rank}(A)=n\iff \text{rank... | {
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Why schemes are $(X,\mathcal O_X)$ rather than $(\mathcal O_X,X)$ or $\{X,\mathcal O_X\}$ Is there a reason why schemes are ordered pairs $(X,\mathcal O_X)$ rather than for example $(\mathcal O_X,X)$ or $\{X,\mathcal O_X\}$?
| This is a general way to talk about structures that consist of several "parts". For example a field is a set $F$ together with two operations ($+$, $\cdot$) and two special elements ($0$ and $1$). So to unambiguosly specify a field, we could denote it with a pentuple $(F,+,\cdot,0,1)$. Then for example a field homomopr... | {
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Show that the set $\{x\in\mathbb{R}^N:\nabla f(x)=0 \}$ is convex Let $f:\mathbb{R}^N\rightarrow \mathbb{R}$ be a $C^1$ convex function. Show that $\{x\in\mathbb{R}^N:\nabla f(x)=0 \}$ is convex (we assume that empty set is convex).
Any hint?
| Convexity implies $f(y)\ge f(x)+\langle \nabla f(x),y-x\rangle$ for all $x,y$. Specializing this to the points of your set, you will find they are points where $f$ attains its global minimum, say $m$. Argue that $\{x:f(x)=m\}$ is convex.
| {
"language": "en",
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What is the required radius of the smaller circles around a larger circle so they touch? I am trying to determine how to calculate the required radius of the smaller circles so they touch each other around the larger circle. (red box)
I would like to be able to adjust the number of smaller circles and the radius of th... | Another approach: Lets say we have $n$ small circles. Then the center points of the small circles form a regular $n$-gon, where the side length is $2r$. The radius of the big circle is the circumradius of the $n$-gon which is $R = \frac{2r}{2\sin(\pi/n)}$ so $r = R \sin(\pi/n)$
| {
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Does Binomial variables independence implies Bernoulli variables independence $X$, $Y$ are independent variables with Binomial distribution. $X={\sum_{i=1}^nX_i}$, $Y={\sum_{i=1}^nY_i}$.
$X_i$, ($1\le i\le n$) are independent Bernoulli variables.
Same applies for $Y_i$
Is the set of $X_i$ and $Y_i$ independent?
| Surprisingly, the answer is no.
Consider the case $n=2$ with probability space $\{0,1\}^4$ and
$X_1, X_2$ the first two coordinate functions and $Y_1, Y_2$ the second two.
The probabilities of the $16$ different configurations are
$$\begin{array}[cccc]{}
x_1 & x_2 & y_1 & y_2 & p(x_1,x_2,y_1,y_2)\cr
0 & 0 & 0 & 0 &1/16... | {
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"url": "https://math.stackexchange.com/questions/1071185",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Can someone help me understand the proof that every cauchy sequence is bounded? This proof is written by a user Batman as an answer to someone's question(just to give credit). Every proof that I've seen is the same idea, and I'm having trouble understanding it intuitively. (I don't see why to take n=N and why the max i... | Just to make things simpler, let's take $\epsilon=1$. Since the sequence is Cauchy, there is an integer N such that $m,n\ge N\implies|a_m-a_n|<1$. In particular, taking $n=N$, we have that $|a_m-a_N|<1$ for $m\ge N$;
so $|a_m|=|(a_m-a_N)+a_N|\le |a_m-a_N|+|a_N|<1+|a_N|$ for $m\ge N$ using the Triangle inequality.
Now... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1071325",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
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Sum of $k$-th powers Given:
$$
P_k(n)=\sum_{i=1}^n i^k
$$
and $P_k(0)=0$, $P_k(x)-P_k(x-1) = x^k$ show that:
$$
P_{k+1}(x)=(k+1) \int^x_0P_k(t) \, dt + C_{k+1} \cdot x
$$
For $C_{k+1}$ constant.
I believe a proof by induction is the way to go here, and have shown the case for $k=0$. This is where I'm stuck. I have look... | Hm. This problem is weird, in that as defined $P_k(n)$ is only defined on the naturals. Although as you noted, you can find a closed form and evaluate it at an arbitrary point.
Have you tried taking the derivative of both sides and using Fundamental Thm of Calculus? That would get rid of the integral, and turns the $C_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1071431",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 3,
"answer_id": 2
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When is a number square in Galois field $p^n$ if it's not square mod $p$? Here is the problem, that I'm stuck on.
There is no square root of $a$ in $\mathbb{Z}_p$. Is there square root of $a$ in $GF(p^n)$?
Well, it's certainly true that $$x^{p^n}=x$$ and $$x^{p^n-1}=1$$ for nonzero $x$.
If there is such an $x$ that $a... | Because finite fields are uniquely determined by their order, you know that if $a\in\Bbb Z$, then one of two things is true, either $a=x^2$ for some other $x\in\Bbb Z/p=\Bbb F_p$, or not. In the case it is not, then $x^2-a$ is irreducible over $\Bbb F_p$. If so then
$$\Bbb F_p[x]/(x^2-a)\cong\Bbb F_{p^2}$$
and this is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1071491",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Finding the radius of a third tangent circle Sorry if this is a foolish question, but I'm having difficulty understanding how to solve for $r_3$ in the following diagram...
According to WolframAlpha's page on tangent circles, the radius of $c_3$ can be calculated using the following formula
$r_3=\frac{r_1 \times r_2}{... | This answer to a slightly different problem
gives a useful diagram showing how to compute the distance between the points
of tangency of two circles and a line, given that the circles are externally tangent
(as yours are).
From this we see that if we label the three points of tangency $A,$ $B,$ and $C$
(in sequence fro... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1071577",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Question on induction technique When one uses induction (say on $n$) to prove something, does it mean the proof holds for all finite values of $n$ or does it always hold when even $n$ takes $\pm\infty$?
| Regular inductive proofs work only over $\mathbb{N}$, the set of natural numbers. The proof that "inductive proofs work" depends on the well-ordering property of $\mathbb{N}$, and $\infty \not\in \mathbb{N}$. As people in the comments have pointed out, induction can be extended to other well-ordered sets as well. Regar... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1071663",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Proving $\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\cdots+\frac{1}{\sqrt{n}}>2-\frac{2}{n}$ by induction for $n\geq 1$ I have the following inequality to prove with induction:
$$P(n): \frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\cdots\frac{1}{\sqrt{n}}>2-\frac{2}{n}, \forall n\in \mathbb{\:N... | Initial comment: Begin by noting that, for all $n\geq 1$, we have that
$$
n(\sqrt{n}-2)+2>0\Longleftrightarrow n\sqrt{n}-2n+2>0\Longleftrightarrow \color{red}{\sqrt{n}>2-\frac{2}{n}}.\tag{1}
$$
Thus, it suffices for us to prove the proposition $P(n)$ for all $n\geq 1$ where
$$
P(n): \frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1071752",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Under what conditions the point $z$ is unique in its range of existence? Theorem: Let $f$ be continuous on $[a,b]$. If the range of $f$ contains $[a,b]$, then the equation $f(x)=x$ has at least one solution $z$ in $[a,b]$, i.e., $f(z)=z$.
My question is:
Under what conditions the point $z$ is unique in its range of exi... | This is a sufficient condition:
$$
|f(x)-f(y|<|x-y|\quad\forall x,y\in[a,b].
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1071850",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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In triangle ABC, Find $\tan(A)$.
In triangle ABC, if $(b+c)^2=a^2+16\triangle$, then find $\tan(A)$ . Where $\triangle$ is the area and a, b , c are the sides of the triangle.
$\implies b^2+c^2-a^2=16\triangle-2bc$
In triangle ABC, $\sin(A)=\frac{2\triangle}{bc}$, and $\cos(A)=\frac{b^2+c^2-a^2}{2bc}$,
$\implies \ta... | HINT:
$$16\triangle=(b+c+a)(b+c-a)$$
$$\iff16rs=2s(b+c-a)$$
$$8r=b+c-a$$
Using this and $a=2R\sin A$ etc.,
$$8\cdot4R\prod\sin\dfrac A2=2R\cdot4\cos\dfrac A2\sin\dfrac B2\sin\dfrac C2$$
$$\implies4\sin\dfrac A2=\cos\dfrac A2$$ as $0<B,C<\pi,\sin\dfrac B2\sin\dfrac C2\ne0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1071953",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
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Interpreting the Möbius function of a poset I have just learned about incidence algebras and Möbius inversion. I know that the Möbius function is the inverse of the zeta function, and that it appears in the important Möbius inversion formula. But does it have any interpretation outside these two contexts? Does the Möbi... | Here is one answer: Every interval in the poset has an associated abstract simplicial complex consisting of all chains in the interval that do not contain the maximum or the minimum. This complex gives rise to some topological space, which is often an interesting space for common posets (e.g. a sphere). The Mobius func... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1072025",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Solving second order difference equations with non-constant coefficients
For the difference equation
$$
2ny_{n+2}+(n^2+1)y_{n+1}-(n+1)^2y_n=0
$$
find one particular solution by guesswork and use reduction of order to deduce the general solution.
So I'm happy with second order difference equations with constant co... | Hint: Write the equation as
$$2n(y_{n+2} - y_{n+1})= (n + 1)^2(y_{n} - y_{n+1})$$
or
$$2n A_{n+1} - (n+1)^2A_{n}$$
with solution
$$A_{n} = (n!)^2/ 2^{(n-1)}(n-1)!$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1072156",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Setting corresponding entries in a matrix I've recently read "Matrix Inversion and the Great Injustice", a rather humorous article of a student venting his frustrations due to feeling as if he has been graded unfairly.
I follow everything so far, up until this part (at the bottom of the first page):
I was asked to sol... | It means that if you have two matrices:
$\begin{bmatrix}
a&b\\c&d
\end{bmatrix}=\begin{bmatrix}
h&i\\j&k
\end{bmatrix}$
then it must be the case that $a=h, b=i, c=j, d=k$.
In your case,
$$\begin{bmatrix}\mathbf{x}_1 + 3\mathbf{x}_3 & \mathbf{x}_2 + 3\mathbf{x}_4\\\mathbf{x}_1 + \mathbf{x}_3 & \mathbf{x}_2 + \mathbf{x}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Putnam definite integral evaluation $\int_0^{\pi/2}\frac{x\sin x\cos x}{\sin^4 x+\cos^4 x}dx$
Evaluate $$\int_0^{\pi/2}\frac{x\sin x\cos x}{\sin^4 x+\cos^4 x}dx$$ Source : Putnam
By the property $\displaystyle \int_0^af(x)\,dx=\int_0^af(a-x)\,dx$:
$$=\int_0^{\pi/2}\frac{(\pi/2-x)\sin x\cos x}{\sin^4 x+\cos^4 x}dx=\fr... | \begin{align}
\int_0^{\pi/2}\frac{\sin x\cos x}{\sin^4x+\cos^4x}\mathrm dx&=\int_0^{\pi/2}\frac{\sin x\cos x}{\sin^4x+\left(1-\sin^2x\right)^2}\mathrm dx\\[7pt]
&=\int_0^{\pi/2}\frac{\sin x\cos x}{2\sin^4x-2\sin^2x+1}\mathrm dx\\[7pt]
&=\frac14\int_0^1\frac{\mathrm dt}{t^2-t+\frac12}\qquad\color{blue}{\implies}\qquad t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1072316",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 4,
"answer_id": 0
} |
Classification of numbers on the base of binary representation The problem is the following. I would like to find a simple algorithm or principle of classification of numbers regarding their presentation in binary form.
Let's consider an example. The numbers by 4-digit binary numbers are following:
$0000_2=0$
$0001_2=1... | The number of ones in the binary representation of $n$ is the greatest integer $r$ such that $2^r$ divides
$$\binom{2n}n$$
See https://oeis.org/A000120
It is not too hard (but not too simple, either) to prove this by induction.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1072393",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
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What are some applications of elementary linear algebra outside of math? I'm TAing linear algebra next quarter, and it strikes me that I only know one example of an application I can present to my students. I'm looking for applications of elementary linear algebra outside of mathematics that I might talk about in discu... | Computer science has a lot of applications!
*
*Manipulating images.
*Machine learning (everywhere).
For example: Multivariate linear regression $(X^TX)^{-1}X^{T}Y$. Where X is an n x m matrix and Y is an N x 1 Vector.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1072459",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "69",
"answer_count": 20,
"answer_id": 7
} |
Questionable Power Series for $1/x$ about $x=0$ WolframAlpha states that
The power series for $1/x$ about $x=0$ is:
$$1/x = \sum_{n=0}^{\infty} (-1)^n(x-1)^n$$
This is supposedly incorrect, isnt it?
This is showing the power series about $x=1$ in the form $(x - c)$
I dont understand how WolFramalpha says that is corr... | It is rather strange, because Wolfram Alpha is perfectly happy to return a Laurent series for e.g.
series of 1/(x+x^2) at x = 0
Somehow, $1/x$ is treated differently.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1072532",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Prove $f:\mathbb{N} \rightarrow \mathbb{R}$ is continuous using the definition of sequential continuity The definition of sequential continuity is that $x_n \rightarrow x \implies f(x_n) \rightarrow f(x)$.
If the terms of the sequence $\{x_n\}$ are only natural numbers, I know that for all $\epsilon > 0$, we can find a... | Any function $f : \mathbb N \to \mathbb R$ is continuous. To show this using sequential continuity, let $\{x_n\}$ be a sequence in $\mathbb N$ that converges to $x$. A convergent sequence in $\mathbb N$ is eventually constant. But then, $\{f(x_n)\}$ is also eventually constant. Say $f(x_n) = f(x)$ for $n > N$ for some ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1072712",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Faulty proof that $V=U_1 \oplus W$ and $V=U_2 \oplus W$ implies $U_1 = U_2$ The question is as follows:
Prove or give a counterexample: if $\ U_1, U_2, W$ are subspaces of $V$ such that
$V=U_1 \oplus W$ and $\ V = U_2 \oplus W$,
then $\ U_1 = U_2$.
I happily proved this but then found out that it is in fact incorrect... | $u\in U + W$ does not mean that $u\in U$ or $w\in W$; rather, it means that $u$ is the sum of something in $U$ and something in $W$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1072779",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Prove that ${x dy - y d x\over x^2 + y^2}$ is not exact Please can someone verify my proof that
$$\psi = {x dy - y d x\over x^2 + y^2}$$
is not exact?
Here is my work:
If $\psi$ was exact there would exist $f:\mathbb R^2 \setminus \{0\} \to \mathbb R$ such that $df = f_x dx + f_y dy = \psi$. Here $f_x = {-y \over x^2 +... | By Green's theorem,
$$\int_{S^1} \psi = \int_{S^1} x\, dy - y\, dx = \iint_{D^2} \text{div}(\langle x,y\rangle)\, dA = 2\cdot \text{Area}(D^2) = 2\pi$$
Alternatively, parametrize $S^1$ by setting $x = \cos(t)$, $y = \sin(t)$, $0 \le t \le 2\pi$. Then $$\int_{S^1} \psi = \int_0^{2\pi} (\cos(t)\cdot \cos(t) - \sin(t)\cd... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1072852",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
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Every simple planar graph with $\delta\geq 3$ has an adjacent pair with $\deg(u)+\deg(v)\leq 13$
Claim: Every simple planar graph with minimum degree at least three has an edge $uv$ such that $\deg(u) + \deg(v)\leq 13$. Furthermore, there exists an example showing that $13$ cannot be replaced by $12$.
This seems rel... | This may be better as a simple comment, but I lack the reputation.
It can be assumed that we are working with a triangulation, but we need to be careful about which edges we add when triangulating. Suppose that the graph has minimum degree three and there is no edge $uv$ with $\deg(u) + \deg(v) \leq 13$. Now suppose th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1072921",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 2,
"answer_id": 0
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Galois finite extension Let $K/ \mathbb{Q}$ be a finite Galois extension, $K \otimes_{\mathbb{Q}} \mathbb{R} \simeq \mathbb{R}^s \oplus \mathbb{C}^t$.
Prove that either $s=0$ or $t=0$.
| The automorphisms of $K/\Bbb Q$ are continuous, so they extend to a completion. If we denote an infinite place by $\mathfrak{p}$, we know one completion is $K_{\mathfrak{p}}$. As all other completions are given by $\sigma(K)_{\sigma(\mathfrak{p})}$ and $\sigma(K)=K$ (since we are dealing with a Galois extension) we use... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1073027",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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Integral $\int_{1}^{2011} \frac{\sqrt{x}}{\sqrt{2012 - x} + \sqrt{x}}dx$ Evaluate:
$$\int_{1}^{2011} \frac{\sqrt{x}}{\sqrt{2012 - x} + \sqrt{x}}dx$$
Using real methods only.
I am not sure what to do.
I tried finding a power series, which was too ugly.
I just need some hints, not an answer to do this integral, this is f... | HINT:
As $\displaystyle\int_a^bf(x)\ dx=\int_a^bf(a+b-x)\ dx$
So, if $\int_a^bf(x)\ dx=I,$
$$2I=\int_a^b[f(x)+f(a+b-x)]\ dx$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1073120",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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Inverse Laplace Transformation I have a question about laplace transformation.
$\frac{8s+4}{s^2+23}$
I tried to split them. $\frac{8s}{s^2+23}$ is the image of a cosine and $\frac{4}{s^2+23}$ is the image of a sine.
Here is what I did :
$\frac{8s}{s^2+(\sqrt{23})^2}$ is the image of $8\cos(\sqrt{23}t)$ and $\fr... | If you don't mind some Residue theory, we can check use that to check your solution.
\begin{align}
\mathcal{L}^{-1}\biggl\{\frac{8s+4}{s^2+23}\biggr\}&=\frac{1}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty}\frac{8s+4}{s^2+23}e^{st}ds\\
&=\sum\text{Res}
\end{align}
The poles in the $s$-plane occur at $s=\pm i\sqrt{23}$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1073214",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Undoing anonymous donations All the students in a class are planning to do a trip. Not all of the students can afford it, and it is considered shameful to reveal their poverty. So it is suggested that anyone can donate anonymously to a fund. If the fund becomes big enough to cover the trip, the trip happens. If not, th... | Yes it is possible if the paiement is done via Paypal / Online paiement.
Bank transactions are validated and done only if the fund limit is reached. So, while the fund is not reached, nobody loose money and everybody stay anonymous.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1073290",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Determinant: Alternative Definitions Reference
Foundation for: Determinant: Continuity
Problem
Given a vector space $V$.
Consider an endomorphism $T:V\to V$.
The rank of an endomorphism:
$$\mathrm{rank}T:=\dim\left(\mathrm{im}T\right)$$
The determinant of an endomorphism:
$$\det T:=\text{???}$$
What would be a nice def... | Let $V$ be an $n$-dimensional vector space over the field $\mathbb{F}$.
Given a linear map $T : V \to V$, there is an induced linear map $\bigwedge^nT : \bigwedge^n V \to \bigwedge^n V$ given by $\left(\bigwedge^nT\right)(v_1\wedge\dots\wedge v_n) = (Tv_1)\wedge\dots\wedge(Tv_n)$. As $\bigwedge^nV$ is one-dimensional,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1073376",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 2
} |
Is split-complex $j=i+2\epsilon$? In matrix representation
imaginary unit
$$i=\begin{pmatrix}0 & -1 \\ 1 & 0 \end{pmatrix}$$
dual numbers unit
$$\epsilon=\begin{pmatrix}0 & 1 \\ 0 & 0 \end{pmatrix}$$
split-complex unit
$$j=\begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix}$$
Given this definition, does not it follow that
$$... | If, ignoring the means by which you reached this conclusion (which was well-addressed in epimorphic's answer), we supposed that
$$j=i+2\varepsilon$$
then it follows that (assuming commutativity)
$$j^2=(i+2\varepsilon)^2=i^2 + 4\varepsilon i+4\varepsilon^2$$
which, replacing each by the definition of their square:
$$1=-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1073486",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
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Is it true that the relation |A| < |B| is a sufficient condition for claiming that $f$ is a bijection? This is an exercise of an assignment I have:
Suppose $A$ and $B$ are finite sets and $f\colon A\to B$ is surjective. Is it
true that the relation “$|A| < |B|$” is a sufficient condition for
claiming that $f$ is a... | If $\lvert A \rvert < \lvert B \rvert$, then you cannot have any surjective function $f\colon A\to B$ anyway, and the question is vacuous.
(the image $f(A)$ of $A$ by any function $f$ must satisfy $\lvert f(A) \rvert \leq \lvert A \rvert$, with equality when $f$ is injective).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1073556",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Evaluating $\int_0^{2} \frac{dx}{\sqrt[3]{2x^2-x^3}}$ How to calculate this integral?
$$\int_0^{2} \frac{dx}{\sqrt[3]{2x^2-x^3}}$$
I suppose that it should be parted like this: $$\int_0^{1} \frac{dx}{\sqrt[3]{2x^2-x^3}} + \int_1^{2} \frac{dx}{\sqrt[3]{2x^2-x^3}}$$ but I have no idea how to calculate these two.
Thank yo... | $\sqrt[3]{2x^2-x^3} = x\sqrt[3]{\dfrac{2}{x}-1} \to u = \sqrt[3]{\dfrac{2}{x} - 1} \to u^3 = \dfrac{2}{x} - 1 \to x = \dfrac{2}{u^3+1}$. Can you take it from here?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1073638",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Normal subgroups in groups of odd order I put the following question in my first-year algebra final this year: Suppose $G$ is a finite group of odd order and $N$ is a normal subgroup of order $5$. Show that $N\le Z(G)$. (By the way, this problem has been posed on this site before.)
The proof that I guided them through ... | Let $N$ be a cyclic group of prime order, and let $A$ be any group of automorphisms of $N$. Then the orbits of $N\setminus\{e\}$ under $A$ all have the same size. (The proof is easy, based on the fact that every element of $N\setminus\{e\}$ is a generator of $N$, and hence every element of $N\setminus\{e\}$ is a power ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1073736",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 3,
"answer_id": 1
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Partial Integral of an ellipse this is my first question on stack exchange so please bear with me.
I am trying to generate a synthetic image of an ellipse in Matlab where each pixel is shaded according to how much of that pixel is contained within the ellipse. I had a similar problem for a circle and tackled it by s... | So I know this question is really old but I just wanted to say that I ended up going with a very simple numerical approximation of the integral by approximating the ellipse with 2 line segments in each pixel (that is I basically did a first order trapezoidal approximation). It's not exactly what I wanted but it was ac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1073834",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to simplify $\det(M)=\det(A^T A)$ for rectangular $A=BC$, with square diagonal $B$ and rectangular $C$ with orthonormal columns? Assume a real, square, symmetric, invertible $n \times n$ matrix $M$ and a real, rectangular $m \times n$ matrix $A$ such that $m \geq n$ and $M = A^T A$. Also assume that $A = B C$, wher... | The simplest thing I can think of is to take the QR decomposition of $A=BC$, then $\det(M)$ is simply the square of the product of the diagonal elements of $R$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1073905",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How does $\log(x^2 + 1)$ become $\log(2x^2)$? My textbook attempts to take the big O of $\log(x^2 +1)$. It proceeds by saying $x^2 + 1 \le 2x^2$ when $x \ge 1$. But I don't know how it came up with this idea.
Question:
Why set $x^2+1$ to a random value to be $2x^2$? Why $2$ of all numbers? Why not $x^2$ or $x^3$?
| I can give some observations.
*
*My guess for changing $x^2+1$ to $2x^2$ is to get rid of the $+1$ and get a monomial (one term) so that taking the log is easier.
*$x^3$ would not be good because it is asymptotically too fast compared to $x^2+1$. Try following the textbook using $x^3$ instead of $2x^2$ and you'll g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1074010",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Using Mean Value Theorem to Prove Derivative Greater than Zero I'm working on a problem where at one point I have to show that for $x\ge a$,
$$g (x) = \int_a^x f - (x-a) f \left({a+x \over 2} \right)$$, $g'(x) \ge 0$.
Additional information: I know that $f''(x)\gt0$, $f'(x)<0$, and $f(x)\gt0$.
Here is what I have s... | if $f''(x)>0,x\in [a,b]$, then we know
$$\dfrac{1}{b-a}\int_{a}^{b}f(x)dx\ge f\left(\dfrac{a+b}{2}\right)\tag{1}$$
I think you want prove this well know inequality?
take $b\to x$,then
$$\int_{a}^{x}f(t)dt-(x-a)f\left(\dfrac{x+a}{2}\right)\ge 0$$
Indeed.for $(1)$ inequality we can use
$$f(x)\ge f\left(\dfrac{a+b}{2}\ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1074096",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Convergence of series of $1/n^x$ - pointwise and uniformly, Consider the series $$\zeta(x) = \sum_{n\ge 1}\frac {1}{n^x}.$$
For which $x \in[0,\infty)$ does it converge pointwise? On which intervals of $[0,\infty)$ does it converge uniformly?
My work:
I think that I can state (without proof, since this goes back to Ca... | The given series is the zeta Riemann series and it's pointwise convergent on $(1,+\infty)$. Moreover for all $a>1$ we have
$$\frac{1}{n^x}\le \frac1{n^a},\quad \forall x\ge a$$
and since $\sum\frac1{n^a}$ is convergent then we have uniform convergence on every interval $[a,\infty)$. The given series isn't uniformly con... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1074129",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Comparison of the consequences of uniform convergence between the real and complex variable cases, In the real variable case, I think that uniform convergence preserves continuity and integrability, i.e., for an integral of a sequence of continuous (or integrable) functions, which converge uniformly to some function ov... | You need stronger conditions than uniform convergence to ensure that
$$f_n(x) \to f(x) \implies f_n'(x) \to f'(x).$$
Here is a standard theorem found in virtually all real analysis books.
Suppose $(f_n)$ is a sequence of differentiable functions that converges pointwise at some point in $[a,b]$ and $(f_n')$ converges... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1074205",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
What is $\tau(A)$ of components of $G \backslash A$, where $A \subseteq V$? A graph is $t$-tough if for all cutsets $A$ we have :
definition of t-tough can be found here
http://personal.stevens.edu/~dbauer/pdf/dmn04f6.pdf
Now I am reading a paper which author defines t-tough graph in other terms:
Link to the paper:
h... | From the context you provided, it seems that the definition of $\tau(A)$ is given in the sentence you quote:
$\tau(A)$ is the number of connected components of the graph $G\setminus A$, obtained by removing from $G$ a subset $A$ of its vertices.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1074286",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
periodic solution of $x''-\ (1-\ x^2-\ (x')^2)\ x'+x=0$ Assume differential equation $$x''-\ (1-\ x^2-\ (x')^2)\ x'+x=0$$
I want to discusse about non-constant periodic solution of it.
Can someone give a hint that how to start to think. And does it have periodic solution.
My tries: I changed it to system of differentia... | In terms of $$Z = x^2 + x'^2 - 1$$ the equation becomes
$$Z' = -2x'^2Z$$
which has the solution $Z(t) = Z(0)e^{-2\int_0^t x'^2 dt}$. Now if $x$ is periodic then $Z$ must be periodic, but this is only possible (since $\int_0^t x'^2 dt$ is an increasing function) if $x'\equiv 0$ or $Z(0) = 0$. The only non-constant perio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1074387",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
How many different (circular) garlands can be made using $3$ white flowers and $6m$ red flowers? This is my first question here.
I'm given $3$ white flowers and $6m$ red flowers, for some $m \in \mathbb{N}$. I want to make a circular garland using all of the flowers. Two garlands are considered the same if they can be... | First I will dramatically overcount. Then I will overcompensate for my overcounting. Then I will compensate for my overcompensation to reach the final answer.
Imagine the garland as a fixed circle, with a total of $3+6m$ positions for flowers. Then the number of possible garlands is simply the number of ways to choos... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1074469",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 2
} |
If we have $f(x)=e^x$, then what is the maximum value of $δ$ such that $|f(x)-1|< 0.1$ whenever $|x|<δ$? If we have $f(x)=e^x$, then what is the maximum value of $δ$ such that $|f(x)-1|< 0.1$ whenever $|x|<δ$?
I tried to solve this problem with delta-epsilon definition
from the definition, 1 is L (the value of the limi... | You are almost done.
Your last inequality is
$$\ln\frac9{10}<x<\ln\frac{11}{10}$$
or
$$-\ln\frac{10}9<x<\ln\frac{11}{10}$$
Since $11/10<10/9$, for $|x|<\delta=\ln(11/10)$, the inequality $|f(x)-1|<0.1$ holds. What happens if $|x|\ge\ln(11/10)$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1074546",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Determine monotone intervals of a function Let $$ f(x) = \int_1^{x^2} (x^2 - t) e^{-t^2}dt. $$ We need to determine monotone intervals of $f(x)$. I tried to differentiate $f(x)$ as follows.
$$ f'(x) = \left(x^2 \int_1^{x^2} e^{-t^2}dt \right)' - \left(\int_1^{x^2} te^{-t^2}dt\right)' \\
= 2x \int_1^{x^2} e^{-t^2}dt + ... | Note that $f'(x)=0 \iff x \in \{ -1, 0, 1\} $. It suffices to show that $ \displaystyle\int_{1}^{x^2} e^{-t^2} \, \mathrm{d}t $ is positive for $|x|>1$ and negative for $|x|<1$. Do you see why this is?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1074624",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
how many ways to choose 3 coins? Sorry I don't know the correct math terms here, I haven't had a math class in some time. That's probably why I have trouble finding an existing question like this, too. Let's say there are 4 differnt kinds of coins: penny (P), nickle (N), dime (D), and quarter (Q). How many ways can you... | Here's a solution using a much more general method, the Pólya-Burnside lemma.
We consider the three choices of coins as a single object with three slots that must be filled. The three slots are indistinguishable, so any permutation of them is considered a symmetry of this object; its symmetry group is therefore $S_3$,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1074717",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
In projective geometry the dual of the cross ratio dual is an angle measurement? I am trying to get my head around angles in projective geometry.
I understand (more or less) the cross ratio and that it can be seen as an distance measurement. (for example in the Beltrami Cayley Klein model of hyperbolic geometry)
But th... | You can define angles in terms of the cross ratio (of lines) and the circular points at infinity.
If you have two lines that intersect at the point P, you can draw two more lines from P to the circular points at infinity. Taking the natural logarithm of this will give you the Euclidean angle between the two original li... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1074818",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Jamie rolls a die multiple times. find the probability that she rolls her first 5 before she rolls her second even number Jamie rolls her fair 6-sided die multiple times.
Find the probability that she rolls her first 5 before she rolls her second (not necessarily distinct) even number?
This is what I have so far...
th... | Consider the following two events:
*
*$A:=\left\{\mbox{sequence of rolls containing one or no even number, and ending in a 5}\right\}$
*$B:=\left\{\mbox{sequence of rolls containing some even number}\right\}$
The probability of interest, in a sequence of independent dice rolls, is
$$
P\left(A \mbox{ followed by }... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1074902",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 3
} |
Given $n$ linear functionals $f_k(x_1,\dotsc,x_n) = \sum_{j=1}^n (k-j)x_j$, what is the dimension of the subspace they annihilate?
Let $F$ be a subfield of the complex numbers. We define $n$ linear functionals on $F^n$ ($n \geq 2$) by $f_k(x_1, \dotsc, x_n) = \sum_{j=1}^n (k-j) x_j$, $1 \leq k \leq n$. What is the dim... | All the $f_k$ are linear combinations of the two linear functionals
$$
\sum_{j=1}^n x_j \quad\text{and}\quad \sum_{j=1}^n jx_j;
$$
therefore the dimension is at most $2$. Checking that the dimension is at least $2$ should be easy.
(For an exercise, you might want to use this observation to construct a solution along th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1074979",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
Find $\lim_{x\to 0^+}\sin(x)\ln(x)$ Find $\lim_{x\to 0^+}\sin(x)\ln(x)$
By using l'Hôpital rule: because we will get $0\times\infty$ when we substitute, I rewrote it as:
$$\lim_{x\to0^+}\dfrac{\sin(x)}{\dfrac1{\ln(x)}}$$
to get the form $\dfrac 00$
Then I differentiated the numerator and denominator and I got:
$$\dfrac... | We can use approximation arguments : when $x$ is small $\sin(x) \approx x$ and any polynomial grows faster than logarithm. Hence $\lim_{x \to 0^+} \sin(x) \ln(x) = \lim_{x \to 0^+} x = 0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1075036",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
} |
Triangle-free graph with 5 vertices What is the maximum number of edges in a triangle-free graph on 5 vertices?
No answers, please...just hints.
I believe that E $\leq$ 5, but I'm not sure where to go from there.
| Consider a pentagon. If you try to add any more edges to the pentagon a triangle will be formed. Thus for graph having a cycle containing 5 vertices ( all vertices that is ) can have at maximum 5 edges without violating the condition.
Now consider bi-partite graphs with a total of 5 vertices , say $x$ in one group and ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1075117",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
How find this diophantine equation $(3x-1)^2+2=(2y^2-4y)^2+y(2y-1)^2-6y$ integer solution Find this following Diophantine equation all integer solution
$$(3x-1)^2+2=(2y^2-4y)^2+y(2y-1)^2-6y$$
or
$$9x^2-6x+3=4y^4-12y^3+12y^2-5y$$
Maybe this equation can be solved by using Pell equation methods?
I want take right is Qua... | You can write it as
$$(3x-1)^2-(2y^2-3y+\tfrac34)^2=-\tfrac12y-\tfrac{41}{16}.$$
Factoring the LHS gives two factors at least one of which gets too large as $y$ is large, as
$$|(3x-1)-(2y^2-3y+\tfrac34)|+|(3x-1)+(2y^2-3y+\tfrac34)|\geqslant2\cdot|2y^2-3y+\tfrac34|.$$
It suffices to check the $y$'s with $-\tfrac12y-\tfr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1075179",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Distinct integers with $a=\text{lcm}(|a-b|,|a-c|)$ and permutations Do there exist three pairwise different integers $a,b,c$ such that $$a=\text{lcm}(|a-b|,|a-c|), b=\text{lcm}(|b-a|,|b-c|), c=\text{lcm}(|c-a|,|c-b|)?$$
None of the integers can be $0$, because the lcm is never $0$. So we know that $|a-b|<\max(a,b)$ (an... | It seems the following.
All greatest common divisors considered below are positive. Put $d=\text{gcd}(a,b,c)$. Let $d’=\text{gcd}(a-b,a-c)$ Then $d|d’$. From the other side, $d’|a$, so $d’|b=a-(a-b)$ and $d’|c=a-(a-c)$. So $d’|d$ an therefore $d’=d$. Hence $|a|=|a-b||a-c|/d$. Similarly, $|b|=|b-a||b-c|/d$ and $|c|=|c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1075231",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
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