Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Long Inequality problem for $a, b, c $ positive real numbers $$ \left( a+\frac{1}{b} -1\right) \left( b+\frac{1}{c} - 1\right) +\left( b+\frac{1}{c} -1\right) \left( c+\frac{1}{a} -1\right) +\left( c+\frac{1}{a} -1\right) \left( a+\frac{1}{b} -1\right) \geq 3$$
How we can prove the inequality above. Actually it take... | Hint:
$a,b,c$ are positive real numbers.
So,$a+\frac 1b>0$
So,$a+\frac 1b-1>-1$.
Similarly all the values are greater than $-1$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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How to prove indirectly that if $42^n - 1$ is prime then n is odd? I'm struggling to prove the following statement:
If $42^n - 1$ is prime, then $n$ must be odd.
I'm trying to prove this indirectly, via the equivalent contrapositive statement, i.e. that if $n$ is even, then $42^n - 1$ is not prime.
By definition, for... | Note that
$$42^{2k}-1=(42^k)^2-1=(42^k-1)(42^k+1)$$
where $1\lt 42^k-1\lt 42^k+1$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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"answer_id": 2
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Find the last two digits of the given number Problem:
Find the last $2$ digits of $7^{2008}$.
Unfortunately I have no idea how to solve this problem. I know that for the last digit, we have to apply the concept of cyclicity, but I'm not aware of how to extend this to the last $2$ digits. I would be truly grateful for... | Hint: $~7^{2008}=49^{1004}=(50-1)^{1004}.~$ Now expand using the binomial theorem, and notice that all terms except for the first two are multiples of $50^2$, and therefore of $100$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1456589",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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Derivation of Green's function in Evans' PDE book. In the book of Evans, on page 34 equation $(25)$ isn't the RHS should be minus what is written there, I mean he uses the fact that $\Delta \Phi(y-x) = \delta(y-x)$ on $U$, and he moves the second term in eq. $(24)$ to the RHS.
If this is not the case then how did he de... | The following is Theorem 1 at page 23 in Evans' book
Let $u = \Phi * f$, then $u \in C^2$ and $\color{red}{-}\Delta u = f.$
Then one adopts the notation $$-\Delta \Phi = \delta_0,$$ thanks to which we can formally compute $$-\Delta u = (-\Delta \Phi) * f = \int \delta(x - y)f(y) = f(x).$$
This should fix your sign pr... | {
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The proof of Ramsey's Theorem I try to understand the proof of Ramsey's Theorem for the two color case. There are still some ambiguities.
It says $R(r-1,s)$ and $R(r,s-1)$ exists by the inductive hypothesis. I know the principle of mathematical induction, but I still don't see it.
Furthermore it says in the proof that ... | Let me try to answer the first question.
The inductive hypothesis is $R(r,s)$ exists.
We know $\forall n\in N, R(n,1)=R(1,n)=1$.
Assume $\forall r<r_0, s<s_0$, $R(r,s)$ exists. (induction hypothesis)
Then we want to show $R(r_0,s_0)$ exists.
Then we apply the "Proof for Two Colors" to show that $R(r_0,s_0)≤R(r_0−1,s_0)... | {
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Tile a 1 x n walkway with 4 different types of tiles... Suppose you are trying to tile a 1 x n walkway with 4 different types of tiles: a red 1 x 1 tile, a blue 1 x 1 tile, a white 1 x 1 tile, and a black 2 x 1 tile
a. Set up and explain a recurrence relation for the number of different tilings for a sidewalk of lengt... | Call the number of tilings of length $n$ $t_n$, then to get a tiling of length $n$, you take one of length $n - 1$ and add a red, a white or a blue tile (3 ways); add a black tile to one of length $n - 2$. I.e.:
$\begin{equation*}
t_{n + 2}
= 3 t_{n + 1} + t_n
\end{equation*}$
Directly we find $t_0 = 1$, $t_1 = 3... | {
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Elementary theorems that require AC It seems that AC is hiding (maybe concealed?) even in some elementary results.
An example:
Theorem: Let $X \subseteq \mathbb R$ and let $x_0 \in \mathbb R$ be an accumulation point of $X$. Then there exists a sequence $ \{ a_n \}_{n=1}^\infty $ S.T. $ \{ a_n \} \subseteq X$ and $a... | Yes, this proof uses countable choice. In an essential way, too. It is consistent (without choice) that there is a dense set if reals without a countably infinite subset. In particular every convergent sequence from that set must be eventually constant. But density means that every real is in the closure.
Other proofs ... | {
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"timestamp": "2023-03-29T00:00:00",
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How do I calculate $\lim_{x\to+\infty}\sqrt{x+a}-\sqrt{x}$? I've seen a handful of exercises like this:
$$\lim_{x\to+\infty}(\sqrt{x+a}-\sqrt{x})$$
I've never worked with limits to infinity when there is some arbitrary number $a$. I am not given any details about it.
Apparently the answer is $0$. How was that conclusio... | Notice, $$\lim_{x\to \infty}(\sqrt{x+a}-\sqrt x)$$
$$=\lim_{x\to \infty}(\sqrt{x+a}-\sqrt x)\frac{(\sqrt{x+a}+\sqrt x)}{(\sqrt{x+a}+\sqrt x)}$$
$$=\lim_{x\to \infty}\frac{x+a-x}{\sqrt{x+a}+\sqrt x}$$
$$=a\lim_{x\to \infty}\frac{1}{\sqrt{x+a}+\sqrt x}=a(0)=0$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Find the sinusoidal equation given only a high and a low Researchers find a creature from an alien planet. Its body temperature varies sinusoidally with time. It reaches a high of $120^o F$ in $35$ minutes. It reaches a low of $104^{o}F$ in $55$ minutes.
a) Sketch a graph
b) Write an equation expressing temperature in... | Your reasoning that the period is $40$ minutes is correct. Your equation is not correct. Note that the sine ranges from $-1$ to $+1$, so the multiplier of the sine function is half the peak-to-peak range. The sine averages to $0$, so the constant should be halfway between the maximum and minimum. You also need a ti... | {
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"timestamp": "2023-03-29T00:00:00",
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Is it possible to do summation of a derivative? For example, let's say you want to add a sequence of terms, each succeeding term being the derivative of the previous.
ex: the term is x4, and you want to add 4 derivatives of it in a sequence of terms.( = 4x3+12x2+24x+24 )
What would the notation for this be? If possible... | Another way to
actually do a summation:
Since
$f(x+h)
=\sum_{n=0}^{\infty} \frac{h^n f^{(n)}(x)}{n!}
$,
by setting
$h=1$
we get
$f(x+1)
=\sum_{n=0}^{\infty} \frac{f^{(n)}(x)}{n!}
$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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For complex z showing $\cos(z+ \pi) = -\cos(z)$ As the title states, for complex $z$ I want to show $\cos(z+ \pi) = -\cos(z)$.
My first attempt was to change $\cos$ into $(e^{iz} + e^{-iz}) /2$ but then I figured using the identity $\cos(z) = \cos(x)\cosh(y)+i\sin(x)\sinh(y)$ was better since $\cos(x+ \pi)=-\cos(x)$ fo... | The sleek, more complex-analytic way to do it: Let $f(z) = \cos z + \cos(z+\pi)$. Then $f$ is entire, and $f(x) = 0$ for $x \in \mathbb{R}$. Hence, by the identity theorem, $f(z) = 0$ for all $z \in \mathbb{C}$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Let $M$ be a finitely generated $R$-module and $I \subset R$ an ideal such that $IM = M$. If $M'$ is a particular submodule, does $IM '= M'$?
Suppose $R$ is a commutative ring, $I$ an ideal, and $M$ a finitely generated $R$-module with generators $\{m_1, \ldots, m_n\}$; suppose further that $IM = M$. Let $M'$ be the s... | Since $M=IM$, we can write $m_1=a_1m_1+a_2m_2+\dots+a_nm_n$, for $a_i\in I$. Rearranging terms, we find $(1-a_1)m_1\in IM'$. Now use this to show that $(1-a_1)m_i\in IM'$ for $i=2,\dots,n$ as well. Since $a_1\in I$, conclude that in fact $m_i\in IM'$ for $i=2,\dots,n$.
| {
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Is this equality about derivative of a polynomial valid? Why is $\left(x^2-1\right)\frac{d}{dx}\left(x^2-1\right)^n = 2nx\left(x^2-1\right)^{n-1}$? This is in a textbook and says that its proof is left as an exercise. It seems to be a difficult identity.
I believe this should just be $\left(x^2-1\right)\frac{d}{dx}\lef... | Just to summarize what various commenters said, the chain rule tells us that
$$(f^n)'=nf^{n-1}f'.$$
If we let $f(x)=x^2-1$, we can see that
$$\frac{d}{dx}\left[\left(x^2-1\right)^n\right]=n\left(x^2-1\right)^{n-1}(2x).$$
Multiplying both sides of the above equation by $x^2-1$, we have, as you correctly assumed:
$$\begi... | {
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Spectral Decomposition of A and B. I was given the following question in my linear algebra course.
Let $A$ be a symmetric matrix, $c >0$, and $B=cA$, find the relationship between the spectral decompositions of $A$ and $B$.
From what I understand. If $A$ is a symmetric matrix, then $A=A^T$. A symmetric matrix has $n$... | Yes, I would say that you have fully described the relationship between $A$ and $B$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finding a position vector I am given position vectors: $\vec{OA} = i - 3j$ and $\vec{OC}=3i-j$.
And asked to find a position vector of the point that divides the line $\vec{AC}$ in the ratio $-2:3$.
So I found the vector $\vec{AC}$, and it is $2i+2j$. Then, if the point of interest is $L$, position vector $\vec{OL} = ... | Let the position vector of the point say $D$ be $\vec{OD}=ai+bj$ then we have
$$\vec{AD}=\vec{OD}-\vec{OA}=ai+bj-(i-3j)=(a-1)i+(b+3)j$$
$$\implies |\vec{AD}|=\sqrt{(a-1)^2+(b+3)^2}$$
$$\vec{CD}=\vec{OD}-\vec{OC}=ai+bj-(3i-j)=(a-3)i+(b+1)j$$
$$\implies |\vec{CD}|=\sqrt{(a-3)^2+(b+1)^2}$$
Now, the point $D$ lies on the (... | {
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"timestamp": "2023-03-29T00:00:00",
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The radius of the inscribed sphere. At the base of a triangular pyramid $SABC$ is an isosceles triangle $ABC$, in which $AB = AC = a$ and the angle $BAC = \alpha$. All the sides are tilted to the plane of the base under the same angles and side $AC$ (or $AB$) forms with a lateral edge $SBC$ angle $\beta$. Determine the... | For the lateral faces to form the same dihedral angle with the base it is necessary that the projection $H$ of vertex $S$ onto the base be equidistant from base sides: $HN=HM$ in the figure below, where $N$ is the midpoint of $BC$. If $K$ is the projection of $A$ onto the opposite face $BCS$, then $AK$ and $SH$ meet at... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Confusion with a function transformation I got a HW problem wrong in my Signals and Systems class and am hoping someone can help me understand why.
There's a discrete-time signal x[n] = u[n] + 2u[n-3] - 3u[n-6], where u[n] is the discrete-time unit step. The problem said to first draw this function, then draw the func... | If you write $x[4n+1] = x\left[4\left(n+\frac{1}{4}\right)\right]$, you would see more clearly which is the correct sequence of operations.
To better understand this, let $y_1[n] = \mathcal{T}_1\{x[n]\} = x[4n]$ be the output of a system that downsamples its input by a factor of 4, and $y_2[n] = \mathcal{T}_2\{x[n]\}=x... | {
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Suppose $b,c \in \textbf Z^+$ are relatively prime (i.e., $\gcd(b,c) = 1$), and $a \,|\, (b+c)$. Prove that $\gcd(a,b) = 1$ and $\gcd(a,c) = 1$ Suppose $b,c \in \textbf Z^+$ are relatively prime (i.e., $\gcd(b,c) = 1$), and $a \,|\, (b+c)$. Prove that $\gcd(a,b) = 1$ and $\gcd(a,c) = 1$.
I've been trying to brainstorm... | since $a \,|\, (b+c)$ thus there is $n$ such that $an=b+c$ suppose $\gcd(a,b) = d$ we show that $d=1$.$d \,|\, b$ , $d \,|\, a$ thus $d \,|\, an$ thus $d \,|\, c=an-b$ hence $d \,|\, \gcd(b,c)=1$ so $d=1$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that $\sum\limits_pa_p$ converges iff $\sum\limits_{n}\frac{a_n}{\log n}$ converges I am going through A. J. Hildebrand's lecture notes on Introduction to Analytic Number Theory. I'm currently stuck at the exercises at the end of Chapter 3 (Distribution of Primes I - Elementary Results). The problem statement is:
... | Denote $a=\lim a_n$. If $a\ne 0$, then obviously both series diverge.
So let further $a=0$. Then $a_n=b_n+b_{n+1}+\ldots$, where $b_n=a_n-a_{n+1}\geqslant 0$. We have $$\sum_p a_p=\sum_p (b_p+b_{p+1}+\ldots)=\sum_n \pi(n) b_n.$$)
Next, $$\sum \frac{a_n}{\log n}=\sum_n \left(\sum_{k\leqslant n} \frac1{\log k}\right)b_n.... | {
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Dimension of an affine cone without one variable is equal to the dimension of the projective algebraic set
Let $A:=V(F_1,...,F_k)\subset\mathbb{P}^n$ with $F_j\in k[X_0,...,X_n]$, a projective algebraic set. Let $C(A)\subset \mathbb{A}^{n+1}$ the affine cone over $X$. Show that $\dim A=\dim B$, where $B$ is the cone $... | The cone over a projective variety $A$ of dimension $n$ will have dimension $n+1$. One way to convince yourself of this is as follows: the natural map $\pi \colon C(A) \setminus \{0\} \to A$ is surjective, and for every $p \in A$, $\pi^{-1}(\{p\})$ is a line (missing a point) in $\mathbb A^{n+1}$ which is a variety of ... | {
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"url": "https://math.stackexchange.com/questions/1458632",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Mathematical Difference between "there is one" and "there is EXACTLY one" I know that I can say ∃x(P(x)) which means there is at least one x for P(x), but how do I express for exactly one?
Here's the questions:
(a) Not everyone in your class has an internet connection.
(b) Everyone except one student in your class has ... | You are correct that "There exists ..." means that there exists at least one. To say that there is exactly one you need to say the following:
$$\exists x(\varphi(x)\land\forall z(\varphi(z)\rightarrow z=x)).$$
Namely, there exists $x$ satisfying whatever, and whenever $z$ satisfies whatever, $z$ has to be equal to $x$.... | {
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How would you interpret this unit conversion question? The following question is copied word for word from my textbook, which is what causes me to be so confused about the contradiction that it implies.
The question:
For gases under certain conditions, there is a relationship between
the pressure of the gas, its volum... | Notice that you're also missing a division by moles. The ideal gas law in physics and chemistry is written as
$$ PV=nRT, $$
where $n$ is the number of moles of the substance. Then the calculation works. Apparently your book uses a different convention, which it should specify.
| {
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"timestamp": "2023-03-29T00:00:00",
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If $x^2+y^2+xy=1\;,$ Then minimum and maximum value of $x^3y+xy^3+4\;,$ where $x,y\in \mathbb{R}$
If $x,y\in \mathbb{R}$ and $x^2+y^2+xy=1\;,$ Then Minimum and Maximum value of $x^3y+xy^3+4$
$\bf{My\; Try::} $Given $$x^2+y^2+xy=1\Rightarrow x^2+y^2=1-xy\geq 0$$
So we get $$xy\leq 1\;\;\forall x\in \mathbb{R}$$
and ... | Using your second last line,
$$f(x,y) = \frac{17}{4} - (xy-\frac 12)^2 $$
now let $\displaystyle xy=u$,
$x^2 + y^2 + xy = 1$ becomes $(x+y)^2 = 1+u$
Therefore $x,y$ are roots of the quadratic $k^2 \pm \sqrt(1+u) k + u = 0.$
If $x, y$ are real, discriminant is non negative, solving this gets $\displaystyle u\leq \frac{1... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Upper bound on $\ln(\frac{1}{1-x})$ for $0\leq x\leq 1/2$ Prove that $$\ln\left(\frac{1}{1-x}\right)\leq x+2x^2$$ for $0\leq x\leq 1/2$.
I thought about the Taylor series $\ln(1+x)=x-x^2/2+x^3/3-\ldots$. For small $x$, the values $1+x$ and $1/(1-x)$ are very close to each other, so the inequality should hold since in t... | Using series expansion, you may get better bounds.. E.g. using $\log (1+t) = t-\frac12t^2+\frac13t^3-\cdots \ge t-\frac12t^2$,
$$\log \frac1{1-x} = -\log (1-x) \le x+\frac12x^2 \le x+2x^2$$
| {
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"url": "https://math.stackexchange.com/questions/1459141",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Understanding how Nehari's problem connects with robust stabiliziation and Nevanlinna-Pick I'm reading Young's "An Introduction to Hilbert space". In chapter 15 he writes about robust stabilization in control theory and ends with that this boils down to an interpolation problem called the Nevanlinna-Pick problem. In th... | *
*Nevanlinna-Pick interpolation (as well as Caratheodory-Fejer interpolation) is a special case of Nehari's problem. Indeed, NP interpolation is to find a function $f$ from the unit ball in $H^\infty$ that interpolates the given values $f(\zeta_i)=\omega_i$ in the unit disc. If $L(z)$ is the Lagrange interpolation po... | {
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A non-linear homogeneous diophantine equation of order 3 I'm a math teacher and one of my student have come to me with several questions.
One of them is the following;
Prove that there is no positive integral solution of the equation
$$x^2y^4+4x^2y^2z^2+x^2z^4=x^4y^2+y^2z^4.$$
I have tried several hours but failed.
Hel... | Let $f(x,y,z) = xy(x+y) +z^2(x-y)$. As you already have in the post, we need to prove that $f(x,y,z)=0$ implies $xyz=0$ over integers.
First, rewrite $f(x,y,z) = y x^2 + (y^2+z^2) x - z^2 y$ and consider $f(x,y,z)=0$ as a quadratic equation in $x$. Then the discriminant of the quadratic equation is
$$
D=(y^2 + z^2)^2... | {
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"timestamp": "2023-03-29T00:00:00",
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Does the measurability of $x\mapsto\operatorname P_x[A]$ imply the measurability of $x\mapsto\operatorname E_x[X]$? Let
*
*$(\Omega,\mathcal A)$ and $(E,\mathcal E)$ be measurable spaces
*$(\operatorname P_x)_{x\in E}$ be a family of probability measures on $(\Omega,\mathcal A)$ such that $$E\ni x\mapsto\operatorna... | In addition to the "approximation by simple functions" approach, one can use the monotone class theorem for functions, as found for example here. The conditions of the theorem quoted there are met by taking the $\pi$-system to be your ${\mathcal A}$ and the vector space ${\mathcal H}$ to be the class of bounded ${\ma... | {
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} |
space of finite borel measure is the dual of continuous function vanishing at infinity I have a question. Why the space of finite borel measure is the dual of the space of continuous fucntions that vanish at infinity?
If we have a finite borel measure, then any continuous function vanishing at infinity integrate with ... | The idea you mention in your last paragraph is precisely the idea behind the proof of the Riesz Representation Theorem (Riesz-Markov, or Riesz-Markov-Kakutani depending on the source).
What makes your objection disappear is the boundedness of the functional. For details, you will have to check the proof of the theorem... | {
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"timestamp": "2023-03-29T00:00:00",
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Prove that $(x_1y_1 + x_2y_2 - 1)^2 \ge (x_1^2 + x_2^2 - 1)(y_1^2 + y_2^2 - 1)$ $x_1, y_1, x_2, y_2 \in \mathbb R$
$x_1^2 + x_2^2 \le 1$
Prove that $(x_1y_1 + x_2y_2 - 1)^2 \ge (x_1^2 + x_2^2 - 1)(y_1^2 + y_2^2 - 1)$
I don't know how to start.
| Let us put $d=LHS-RHS$. View $d$ as a quadratic form in $y_1$ and $y_2$. Writing $d$ as a signed sum of squares using standard techniques, we obtain
$$
d=\frac{\bigg(y_2(1-x_1^2)-x_2(1-x_1y_1)\bigg)^2+(1-x_1^2-x_2^2)(y_1-x_1)^2}{1-x_1^2}
$$
which is indeed nonnegative (note that when $1-x_1^2=0$, the RHS is zero).
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Closed set $F$ is the boundary of any subset of $\mathbb{R}^n$ I need show that any closed subset $F\subset\mathbb{R}^n$ is the boundary of some set $A$ in $\mathbb{R}^n$.
Intuition tells me to take $A=F\setminus(\mathbb{Q}^n\cap int(F))$ and $int(F)$ is the set of interior points of $F$ but I can't prove that boundary... | If x is in the boundary of A the it is an adherent point of A and thus an adherent point of F. F is closed so all adherent points of F are in F. So x is in F.
| {
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Continuous Random Variable question, Probability and Statistics a little help please
A couple decide they really want a daughter. So, they decide to start having children and continue until they have their first daughter. Assuming having either a boy or girl is equally likely, answer the following:
(a) In the end, will... | (a) Expectation on the girl is of course $1$. And for the boys we have something like $P\{B=k\}=\frac{1}{2^{k+1}}$, where $k\in\{0,1,...\}$, $B$- distribution that counts boys. $$\mathbb{E}[B]=\sum_{k=0}^\infty \frac{k}{2^{k+1}}=1.$$
It's exactly the $1/2$ that they have alone girl, and $1/2$ that girl+any positive nat... | {
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How can pi have infinite number of digits and never repeat them? I am very confused about this matter, even if I searched google about this already. Please show me how this is determined and/or at least explain to me.
First, I saw this "Infinite Monkey Theorem" that says given infinite number of tries, a monkey could w... | For the first question: I think you are confusing things here. The Infinite Monkey Theorem does state that, but it doesn't state that the monkey should keep writing the same play of Shakespeare's over and over again in some particular pattern. The same thing goes for $\pi$.
For the second question: the thing is that ne... | {
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The Modified Faro shuffle. I came across this problem in George Andrews's book "Number Theory" (section 4-3, problem 2). It deals with a modified version of the Faro shuffle that behaves as such:
$$(1,2,3,4,5,6,7,8) \rightarrow (4,5,3,6,2,7,1,8)$$
Essentially you cut the deck in half, flip the top half ($1,2,3,4$ in o... | This shuffle is known as a 'Milk Shuffle' and its inverse is the 'Monge Shuffle'. See The mathematics of the flip and horseshoe shuffles Section 6. The Monge Shuffle for Two-Power Decks suggests a method for solving this problem.
Ignore the bottom card (since it never changes position) and renumber the cards from botto... | {
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Formal Way to Prove limit without operating on infinity? What is the "formal" way of proving
Limit as x approaches negative infinity of f(x)
Where f(x) = sqrt(5-x)
I know it's positive infinity but in order to get that I had to "operate" on infinity which is not allowed. Is there a different way to do this problem suc... | Show that for any $M>0,$ there is an $N_M<0$ such that for $x<N_M$ we have $f(x)>M.$
This is the (usual) definition of $f(x)$ increasing without bound as $x$ decreases without bound.
Note that "infinity" never came up. The symbol $\infty$ is in large part just notational shorthand for unboundedness.
| {
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Ring isomorphism $\phi:\Bbb Q[x]\to\Bbb Q[x]$ sending $\phi:x\mapsto (y+1)$ The question says:
"Show that the map $\phi:\Bbb Q[x]\to\Bbb Q[x]$ sending $\phi:x\mapsto (y+1)$ is a ring isomorphism."
$y$ is not defined anywhere. My question is, with superior knowledge of these sort of problems, what was meant to be aske... | I think it's a typo, it should be $x\mapsto x+1$, as you said. This is an homomorphism:
if $p(x),q(x)\in \Bbb Q[x]$, then $\phi(p(x)q(x))$ is just the product
$pq$ evaluated in $x+1$, which is the same as $p(x+1)q(x+1)$, i.e., $\phi(p(x))\phi(q(x))$. This is essentially the fact that the evaluation map is a ring h... | {
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How do I symbolically prove that $\lim_{n \to \infty } (n-n^2)=- \infty $? Intuitively we know that $n^2$ grows faster than $n$, thus the difference tends to negative infinity. But I have trouble proving it symbolically because of the indeterminate form $\infty - \infty$. Is there anyway to do this without resorting to... | Another way: complete the square $n-n^2=\frac14-\bigl(n-\frac12\bigr)^2\to -\infty$.
| {
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Why $\mathrm{adj}(A)\cdot A = A\cdot\mathrm{adj}(A)$? I know that $A\cdot\mathrm{adj}(A) = \det(A) \cdot I$, but why $\mathrm{adj}(A)\cdot A = A\cdot\mathrm{adj}(A)$?
| Let $A$ be an $n \times n$ matrix, $A_{i,j}$ the $(i,j)$-minor of $A$ and $C_{i,j}$ the $(i,j)$-cofactor of $A$, defined as:
$$
C_{i,j} = (-1)^{i+j}A_{i,j}.
$$
By definition we know that the adjungate of $A$ is:
$$
\operatorname{adj} A = [C_{j,i}].
$$
The cofactor expansion along rows gives for all $i,j=1,\dots,n$:
$$
... | {
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Proving that a set is nowhere dense. Let $A\subset X$ be dense in $X$. If $E$ is closed in $X$ and $E\cap A = \emptyset$, then I want to prove that $E$ is nowhere dense.
My attempt:
We will prove that $X \setminus \overline{E}=X \setminus E$ is dense in $X$. So we first note that since $A$ is dense in $X$ we have that ... | Try this way, let $x\in E$ be interior point. then $B(x,r)$ for some $r$ positive, is contained in $E$ and hence open and hence must intersect $A$ which is contradiction
| {
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Is it possible to represent triples of numbers from some small set $X$ uniquely using pairs of numbers from $X$? Given that I have a small range of numbers in ascending order with no duplicates, e.g.,
$$23, 24, 25, 26, 27, 28, 29, 30, 31, 32,$$
and three numbers are chosen from this range, let's say $30, 26,$ and $23,$... | In general, no, but we can as long as the "small range" includes no more than $5$ numbers.
Since their quantities don't matter, there's no harm in relabeling the numbers in the set $[n] := \{1, \ldots, n\}$, where $n$ is the number of elements in the set. Now, we can reframe the question as asking for an surjective map... | {
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Calculating probability on sets I was reading about calculating the support and confidence in regard to "associate rule mining" and found the following definitions:
An association rule is defined as: $A \rightarrow B$ where $A\subset T$, $B\subset T$, and $A \cap B = \emptyset$.
Support: $c(A \rightarrow B) = P(A \cup ... | The first answer is the PIE formula for beer or milk, but the question is asking for beer and milk.
The second answer is correct - if you think about it, picking a random transaction, 2 of the 4 transactions have beer and milk, and so the probability is $\frac24$.
As for your notation, $P(\{\text{Beer, Milk}\}=P(\{Beer... | {
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lHopitals $ \lim_{x\rightarrow \infty} \; (\ln x)^{3 x} $? $ \displaystyle \lim_{x\rightarrow \infty} \; (\ln x)^{3 x} =$ ?
Okay, so what do I do with that power? I need to rewrite the term as fractions. How?
If it was the inner function that's in the power of something: $\ln x^{\frac{1}{3 x}}$ then I'd just simply rew... | Why do you hant to use l'hopital ?
$$(\ln x)^{3x}=e^{3x\ln(\ln(x))}$$
and since $3x\ln(\ln(x))\underset{x\to \infty }{\longrightarrow }\infty $,
$$\lim_{x\to\infty }(\ln x)^{3x}=\infty. $$
| {
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Question About Definition of Almost Everywhere I suppose I'm a bit confused about the definition in the following regard:
A property holds a.e. if it holds everywhere except for a set of measure $0$. Now, if the particular property is only defined for a set of measure $0$, is it a.e. by default?
Say I have two 'conti... | Here's an example to show that your concern about isolated points is justified.
Let $f(x)=0$ for all $x\in{\Bbb R}$ and $g(x) = \max(x,0)$. These two functions are continuous and the are equal at (Lebesgue) a.e. point of $B:=(-\infty,0]\cup\{1\}$. But they are not identically equal on $B$.
| {
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Proof of mean of binomial distribution by differentiation The mean of the binomial distribution, i.e. the expectation for number of events, is $np$. I've seen this proven by rearranging terms so that $np$ comes out. For example here, Relating two proofs of binomial distribution mean
I've also seen the following logic:
... | Let $f(x)=(x+q)^n$, where $q=1-p$ is a constant. Then
$$f(x)=\sum_{k=0}^n \binom{n}{k}x^kq^{n-k}.$$
Differentiate with respect to $x$. We get
$$n(x+q)^{n-1}=\sum_{k=0}^n k\binom{n}{k}x^{k-1}q^{n-k}.$$
Multiply through by $x$, and set $x=p$.
Remark: Here is a nicer proof. Let random variable $X_i$ be equal to $1$ if th... | {
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Are we allowed to choose infinite number of elements from an infinite set?
$f$ is surjective $\implies f$ has right inverse.
Suppose $f$ is surjective. Then for any $b \in B$ there's at least one $a \in A$ such that $f(a) = b$. Choose one such $a$ for each $b$ and define $g: B \implies A$ by letting $g(b)$ the chosen... | It's not a big deal or a problem, but it does require the Axiom of Choice.
Here's the point. We want to base our math on set theory. In elementary situations sets are just things that have elements, and the way they work is just the way things with elements obviously work. That's the level at which you'd say what's the... | {
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Why there's no chain rule for integrals of elementary functions which are expressible in terms of elementary functions? The derivative of every elementary function is elementary; this is owing to the existence of the chain rule for differentiation.
On the other hand, the integral of an elementary function may turn out... | Differentiation is a function that satisfies linearity f(x + y) = f(x) + f(y) and f(ax) = af(x). It also satisfies the rule f(xy) = f(x)y + xf(y). Integration can be thought of as the inverse function much like division can be thought of as the inverse function to multiplication. However, just as division has only some... | {
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Stuck solving a logarithmic equation $$\log _{ 2 }{ 2x } =\log _{ 4 }{ 4x^{ 6 } } -\log _{ 2 }{ 2x } $$
Steps I took:
$$\frac { \log _{ 4 }{ 2x } }{ \log _{ 4 }{ 2 } } =\log _{ 4 }{ 4x^{ 6 } } -\frac { \log _{ 4 }{ 2x } }{ \log _{ 4 }{ 2 } } $$
$$2\log _{ 4 }{ 2x } +2\log _{ 4 }{ 2x } =\log _{ 4 }{ 4x^{ 6 } } $$
$$... | $$
\log_{2} 2x = \log_{4} 4x^{6} - \log_{2}2x \quad \text{iff} \quad \frac{\log 2x}{\log 2} = \frac{\log 4x^{6}}{\log 4} - \frac{\log 2x}{\log 2};\\
\frac{\log 2x}{\log 2} = \frac{\log 4x^{6}}{\log 4} - \frac{\log 2x}{\log 2} \quad \text{iff} \quad 2\log 2x = \log 4x^{6} - 2\log 2x;\\
2\log 2x = \log 4x^{6} - 2\log 2x ... | {
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Is there an easier way to solve this logarithmic equation? $$2\log _{ 8 }{ x } =\log _{ 2 }{ x-1 } $$
Steps I took:
$$\frac { \log _{ 2 }{ x^{ 2 } } }{ \log _{ 2 }{ 8 } } =\log _{ 2 }{ x-1 } $$
$$\frac { \log _{ 2 }{ x^{ 2 } } }{ 3 } =\log _{ 2 }{ x-1 } $$
$$\log _{ 2 }{ x^{ 2 } } =3\log _{ 2 }{ x-1 } $$
$$2\log _{ ... | Notice, $\ \ \large \log_{a^n}(b)=\frac{1}{n}\log_a(b)$
Now, we have $$2\log_8x=\log_2x-1$$
$$2\log_{2^3}x=\log_2x-1$$
$$\frac{2}{3}\log_{2}x=\log_2x-1$$ $$\frac{1}{3}\log_{2}x=1$$
$$\log_2x=3\implies x=2^3=\color{red}{8}$$
| {
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How can I find the inverse of $h(x)=-x(x^3 +1)$ How can I find the inverse of $h(x)=-x(x^3+1)$? it's asked also to find $h^{-1}(2)$ and $h^{-1}(-2)$. I think it's easy to find a domain where this function is bijective. I've already find $h^{-1}(-2)=1$. My problem is to find $h^{-1}(2)$ and the inverse itself.
Thanks
| It's complicate to find the inverse of a non bijective function... In particular, your function is neither one to one, nor onto. Indeed, $$h(-1)=h(0)=0$$ and $$h^{-1}(2)=\emptyset,$$
that's the reason that you can't find $h^{-1}(2)$ ;-)
Restriction of the domain and codomain that gives $h$ bijective
$$h'(x)=-(x^3+1)-3x... | {
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The possible number of blue marbles is A boy has a collection of blue and green marbles. The number of blue marbles belong to the sets $\{2,3,4,\ldots,13\}$. If two marbles are chosen simultaneously and at random from this collection, then the probability that they have different colour is $\frac{1}{2}$. The possible n... | Suppose, there are $m$ blue and $n$ green marbles.
There are $\binom{m+n}{2}=\frac{(m+n)(m+n-1)}{2}$ ways to choose $2$ marbles.
There are $mn$ ways to choose $2$ marbles with different colors.
The probability of getting two marbles with different colours is therefore
$$\frac{2mn}{(m+n)(m+n-1)}$$
So, the probability is... | {
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Finding a unit vector when you have two planar vectors and a normal vector. Find a unit vector in the plane of the vectors $A = i + 2j$ and $B = j + 2k$, perpendicular to the vector $C = 2i + j +2k$.
I'm confused as to what the problem is telling me.
I believe this problem is telling me that $A$ and $B$ span a particul... | We note that the vectors $$\vec{a}=(1,2,0) \mbox{ and } \vec{b}=(0,1,2)$$
are linearly independent, and thus span a plane. If you are unfamiliar with these words, I hyperlinked the (possibly) problematic phrases to their respective Wikipedia articles.
Therefore, if we want to find a vector in the plane spanned by $\ve... | {
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$(A \lor B) \to C$ and $(A \to C) \lor (B \to C)$ Which one entails the other? For a homework assignment I have to prove that one of the statements entails the other.
The statements are:
$(A \lor B) \to C$
$(A \to C) \lor (B \to C)$
The only thing that I got so far is either $\lnot(A \lor B) \to C$ or $(A \to C) \land... | HINT: Use the fact that $p\to q$ is equivalent to $\neg p\lor q$. Thus,
$$(A\lor B)\to C)\equiv\neg(A\lor B)\lor C\equiv(\neg A\land\neg B)\lor C\equiv(\neg A\lor C)\land(\neg B\lor C)\;,$$
where I’ve also used De Morgan’s law and distributivity. Now expand use the same fact to convert this to an expression involving $... | {
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What is the meaning of $DX_p$ for $X$ a vector field on a manifold? This is taken from Palis, Geometric Theory of Dynamical Systems, p.55:
Here $X$ is a $C^r$ vector field on $M$. What does the notation $DX_p$ mean?
| This is often called the intrinsic derivative. (This makes sense, more generally, for the section of any vector bundle at a zero.) It is well-defined at a zero of $X$. Think in local coordinates of $X$ as a map from $\Bbb R^n$ to $\Bbb R^n$, and compute its derivative at $0$ (corresponding to $P$). You can check that y... | {
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An example of a series $\sum a_n$ that converges conditionally but $\sum a_n^3$ does not converge Give an example of a series $\sum a_n$ that converges conditionally but $\sum a_n^3$ does not converge conditionally.
I've come up with an example.
$\frac{1}{\sqrt[3]2}-\frac{1}{2\sqrt[3]2}-\frac{1}{2\sqrt[3]2}+\frac{1}{\... | Consider , $a_n=\frac{(-1)^n}{n}$. Then $a_n$ is conditionally convergent. But , $a_n^3=\frac{(-1)^n}{n^3}$ is NOT conditionally convergent ; as it is absolutely convergent.
| {
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Given $v_i∈B^n$, bounding $\sum b_iv_i$ for $b_i= \pm 1$ Let $(v_i)_{i∈ℕ}$ be a vector sequence. Say $(v_i)$ is boundable (under $M$) if there exists a sequence $(b_i)_{i∈ℕ}$ taking values in $\{-1,1\}$ such that $(|\sum^N_i b_iv_i|)_{N∈ℕ}$ is bounded (under $M$). If $(v_i)$ takes values in the unit ball $B^n⊆ℝ^n$, doe... | This answer is based on my misuderstanding of the question, see the second Feryll's comment below.
According to Wojciech Banaszczyk [B1], S. Sevastyanov [S] and, independently I. Bárány (unpublished) proved that $\alpha_n=\sqrt{n}$. Also there is a lot of the generalizations of this inequality for different norms, see... | {
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The number of bits in $N!$ I'm struggling with this homework problem:
If $N$ is an $n$-bit number, how many bits long is $N!$ approximately (in $Θ(·)$ form)?
I know that the number of bits in $N$ is equivalent to log base $2$ of $N$, but that's about it. Any hints on where to start?
| $$\log_2(n!)=\log_2[n\cdot (n-1)\cdots 1]=\log_2 n+\log_2(n-1)+\cdots+\log_2 1$$
There are $n$ terms, and most of the terms are close to $\log_2 n$ in magnitude. Therefore $\log_2(n!)=\Theta(n\log n)$.
(To prove this more formally, use Stirling's approximation.)
| {
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How to represent E(Y|X) in terms of E(X), E(Y), Var(X), Cov(X,Y)? Known things are:
E(Y|X)=aX+b,
Cov(X,Y) exists,
0 < Var(X) < Infinity,
Question:
Represent a,b in terms of E(X), E(Y), Var(X), Cov(X,Y)
I worked out only one equation: E[E(Y|X)]=E(Y)=E(aX+b)=aE(x)+b, i.e. aE(X)+b=E(Y).
But in order to represent a and b ... | Hint: You already worked out half of the answer:
$$E[E(Y|X)]=E(Y)=E(aX+b)=aE(x)+b \implies aE(X)+b=E(Y)$$
Additionally:
$$E [X \, E [Y|X] =E[E[X Y |Y]]=E[XY]=Cov(X,Y)+E[Y][X]$$
but
$$E [X \, E [Y|X]] =E[X (aX+b)]=a E[X^2]+b E[X]=a (Var(X)+E[X]^2)+b E[X]$$
Combining these two equations, and together with the one above... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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The span of a vector space with elements as linear combinations of no more than $r$ vectors has $\dim V \leq r$ If $V=Span \{ \vec{v}_1, \dots, \vec{v}_n \}$ and if every $\{ v_i \}$ is a linear combination of no more than $r$ vectors in $\{ \vec{v}_1, \dots, \vec{v}_r \}$ excluding $\{ v_i \}$, then $\dim V \leq r$
Ho... | If $V = span \{ v_1, \dots, v_n \}$ and if every $\{ v_i \}$ is a linear combination of no more than $r$ vectors in $\{ v_1, \dots, v_r \}$ excluding $\{ v_i \}$, then $dim V \leq r$
Proof: The hypothesis is the same as, for any $v_i$,
$$v_i = \sum_{j=1}^r b_{ij} v_j$$
with $b_{ii} = 0$, in case $i \leq r$ ("excluding ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1463280",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Partial sum of coefficients of polynomials Let me define polynomials of form $1+x^2+x^3+\cdots+x^k$ as $P(k,x)$.
Let $$Q(x)=\prod_{k=1}^{n}P(k,x)$$
How can I find the sum of coefficients for which exponent of $x$ is $\le T$, where $0 \le T \le \frac{n(n+1)}{2}$ (which we define as $S(T,n)$)?
Example for the clarity of ... | I can't help you with a closed formula for $S(T,n)$, but we can construct a fairly simple algorithm for computing this recursively. Defining $a_k^{(n)}$ via $$Q_n(x) = \sum_{k=0}^{\frac{n(n+1)}{2}}a_k^{(n)} x^k$$ then $$Q_{n+1}(x) = Q_n(x)(1+x+\ldots+x^{n+1}) = \sum_{k=0}^{\frac{(n+1)(n+2)}{2}}a_k^{(n+1)} x^k$$
gives t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1463387",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How can I prove this function is discontinuous or continuous similar to Drichlet function? Given a function
$$F(x)= \begin{cases} x^2 & \text{when }x \in \mathbb Q \\3x & \text{when }x \in\mathbb Q^c \end{cases}$$
Show that $F$ is continuous or not on $x=3$ with $\epsilon-\delta$.
I tried to deal with problems just li... | Let $\epsilon > 0$ , be a positive number
To find a $\delta$ such that $|x-3|<\delta$ implies $|f(x)-f(3)|<\epsilon$.
Case-I if $x\in \mathbb Q$
Then $|f(x)-f(3)|=|x^2-9|=|(x-3)(x+3)|$
Therefore we choose our $\delta$ to be $<1$, then $|x-3|<1$ implies $-1<x-3<1$ , adding 6 on both sides we get $
x+3<7$.
$|f(x)-f(3)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1463457",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Do polynomials in two variables always factor in linear terms? Consider a polynomial of one variable over $\Bbb C$:
$$p(x)=a_0+a_1x+\cdots+a_nx^n,\quad a_i\in\Bbb C.$$
We know from the Fundamental Theorem of Algebra that there exists $c,\alpha_i\in\Bbb C$ such that
$$p(x)=c(x-\alpha_1)\cdots(x-\alpha_n),$$
i.e. we can ... | Let me add to Martin's perfect answer that a homogeneous polynomial $f(x,y)$ (=sum of monomials of the same degree) in two variables does factor into linear homogeneous factors in an essentially unique way, that is up to permutations of the factors and multiplication of the factors by constants. More explicitly: $$f(... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
"answer_count": 3,
"answer_id": 2
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Binary multiplication for negative numbers The question is about binary multiplication for negative numbers. Assume we want to multiply -5 * -3 so the result is +15.
1) In the first step, we have to use 2's complement for the inputs.
+5 = 0101 -> -5 = 1011
+3 = 0011 -> -3 = 1101
2) We follow the simple penc... | 1011
* 1101
----------------
[1][1] [1] [1] 1 0 1 1
[0] [0] 0 0 0 0
[1][1] 1 0 1 1
[1] 1 0 1 1 +
1 0 1 1
... 1 1
------------------------------
c7 c6 c5 c4 c3 c2 c1
Does this clarify ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Expected number of tails before the third head I have a question that I am currently modeling on coin tosses. Suppose, a number of coin tosses are done with $p = 1/4$ such that I have a string of tails, then one head, then a string of tails, then another head, and a string of tails and then a third head and so on. Some... | The idea is good. You are letting $X_1$ be the number of tails before the first head, and $X_2$ the number of tails between the first head and the second head, and so on.
We want $E(X_1+\cdots+X_n)$, which by the linearity of expectation is $E(X_1)+\cdots+E(X_n)$.
We have $E(X_i)=\frac{1}{p}-1$, where $p$ is the prob... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Uniqueness for Set in Family of sets This is an exercise in How to prove it by Velleman.
Suppose $\mathcal{F}$ is a family of sets. Prove that there is a unique set $A$ that has the following two properties:
(a) $\mathcal{F} \subseteq \mathcal{P}(A)$
(b) $\forall B(\mathcal{F} \subseteq \mathcal{P}(B) \rightarrow A \su... | Based on your remark that you allready proved existence I preassume that you have proved that $A:=\cup\mathcal F$ satisfies the conditions (a) and b).
Let it be that the set $A'$ also satisfies these conditions. So:
(a') $\mathcal{F}\subseteq\wp\left(A'\right)$
(b') $\forall B\left[\mathcal{F}\subseteq\wp\left(B\right)... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Counting finite zeros among infinite zeros Let $G$ be open in $\mathbb{C}$ and $f:G\rightarrow \mathbb{C}$ be an analytic function.
Let's denote $\deg(f,z)$ to mean the multiplicity of a zero $z$ of $f$, and $Z(f)$ to mean the set of zeros of $f$.
Let $\gamma:[0,1]\rightarrow \mathbb{C}$ be a closed rectifiable curve w... | You can just replace $G$ by $G\setminus g^{-1}(\{0\})$, and now you've reduced to the case where $g$ has no zeroes in $G$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1464039",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to sketch the subset of a complex plane? The question asks to sketch the subset of $\{z\ \epsilon\ C : |Z-1|+|Z+1|=4\}$
Here is my working:
$z=x+yi$
$|x+yi-1| + |x+yi+1|=4$
$\sqrt{ {(x-1)}^2 + y^2} + \sqrt{{(x+1)}^2+y^2}=4$
${ {(x-1)}^2 + y^2} + {{(x+1)}^2+y^2}=16$
$x^2 - 2x+1+y^2+x^2+2x+1+y^2=16$
$2x^2+2y^2+2=... | That's a good question. Unfortunately you can't just square term by term like that. When you write this out, let z=x+iy. If you do that you will get a very complicated algebraic term.
((x-1)^2 + 2*y^2 + (x+1)^2)^2 = 16
As LutzL had brilliantly stated you can make a substitution, in this case a make a parametrization, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1464152",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 3
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Prove $5 \mid (3^{4n} - 1)$ by induction I need to prove by induction that $5 \mid 3^{4n} - 1$ for $n \ge 1$. Base case is true, so now I have to prove that $5 \mid 3^{4(n+1)} - 1$.
I did
$$= 3^{4n+4} -1$$
$$= 3^{4n} 3^{4}-1$$
I guess I need to make a $3^{4n}-1$ appear somewhere to use the inductive hypothesis, but I ... | Without induction:
$$3^{4n}-1=81^n-1^n=(81-1)\left(81^{n-1}+81^{n-2}+\cdots+1\right)$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Does this integral converge? WolframAlpha says that
$$\int_{-\infty}^\infty \int_{-\infty}^\infty \int_{-\infty}^\infty \frac 1{(1+x^2+y^2+z^2)^2} \, dx \, dy \, dz$$
converges, but it cannot compute integrals that are more than three variables.
Does this integral $$\int_{-\infty}^\infty \int_{-\infty}^\infty \int_{-\i... | Let $S(R)$ be the measure of $\{x_1^2+\ldots+x_n^2=R^2\}$. It is obviously $S(1)\cdot R^{n-1}$.
On the other hand:
$$ \int_{\mathbb{R}^n}\frac{1}{(1+\left|x\right|^2)^2}\,dx = \int_{0}^{+\infty}\frac{S(R)}{(1+R^2)^2}\,dR = S(1)\cdot\int_{0}^{+\infty}\frac{R^{n-1}}{(1+R^2)^2}\,dR$$
is convergent only for $n\leq 3$, and:... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Palindromes and LCM A palindrome is a number that is the same when read forwards and backwards, such as $43234$. What is the smallest five-digit palindrome that is divisible by $11$?
I'm probably terrible at math but all I could do was list the multiples out. Any hints for a quicker solution?
| $$\overline{abcba}\equiv 10^4\cdot a+10^3\cdot b+10^2\cdot c+10^1\cdot b+a$$
$$\equiv (-1)^4a+(-1)^3b+(-1)^2c+(-1)^1b+a$$
$$\equiv a-b+c-b+a\equiv 2a-2b+c\equiv 0\pmod{11}$$
To minimize $\overline{abcba}$, let $a=1$ and $b=0$. Then $c\equiv 9\pmod{11}$, so $c=9$. And in fact $\overline{10901}$ works.
| {
"language": "en",
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What is the probability of winning at this game? you throw a fair, 6 sided dice. If the result is 3 or lower, you lose. If not then you can draw the number of cards the dice shows from a standard 52 deck of cards (if you throw a 5 then you draw 5 cards randomly). You win the game if the 4 aces are among the cards you d... | There is a slight error. The final term reads as $\left(\frac{1}{6}\cdot \frac{48\cdot 47}{\binom{52}{6}}\right)$, but where did those numbers actually come from.
$\frac{1}{6}$ because this is the probability that we are in the scenario of drawing six cards.
$\frac{1}{\binom{52}{6}}$ because we are finding probabiliti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1464625",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Packing $8$ identical DVDs into $5$ indistinguishable boxes I am trying to solve this question:
How many ways are there to pack eight identical DVDs into five indistinguishable boxes so that each box contains at least one DVD?
I am very lost at trying to solve this one. My attempt to start this problem involved drawi... | 5 boxes 8 dvds ...
firstly you put one dvd in each box .
and now you solve no. of ways of placing 3 dvds in 5 boxes.
which is same as no of solution to the equation
b1 + b2 + b3 + b4 + b5 = 3 i.e.,
$ (5+3-1)\choose (3)$ = 35 . .... [solution to the equation a1+a2+a3+...an = r is $ (n+r-1) \choose n $ which can... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Intuition behind left and right translations being bijective in a group? In my algebra class, we learn that the maps $l_g(x) = gx$ for $x \in G$ and $r_g(x) = xg$ for $x \in G$ are bijective. The proof given uses the fact that $l_g l_{g^{-1}} = l_{g^{-1}} l_g = 1_G$, so both functions are bijective since $1_G$ is and t... | And no one has pointed out ...
This result is equivalent to the observation that every row and every column of the multiplication table is a permutation of any other row or column. That is, each row and column has all the symbols in it exactly once, so is surjective and injective.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1464844",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 5
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Hitting a roadblock while solving a logarithmic equation $$x^{ 5-\log _{ 3 }{ x } }=9x^2$$
Steps I took:
$$\log _{ 3 }{ x^{ 5-\log _{ 3 }{ x } } } =\log _{ 3 }{ 9x^{ 2 } } $$
$$(5-\log _{ 3 }{ x } )(\log _{ 3 }{ x) } =\log _{ 3 }{ 9x^{ 2 } } $$
$$5\log _{ 3 }{ x } -(\log _{ 3 }{ x } )^{ 2 }=\log _{ 3 }{ 9x^{ 2 } } $$... | Something to try:
Convert your right side to
$$-\log _{ 3 }{ 9x^{ 2 } }=-\log _{ 3 }{( 3x)^{ 2 } }=-2\log_{3}{(3x)}$$
Then you convert your left-side terms to $$\log_{3}{(3x)}$$ instead of $$\log_{3}{(x)}$$
See where that takes you.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show that f(x)=e^x from set of reals to set of reals is not invertible... Yes, this is my question...
How can you prove this? That $f(x)=e^x$ from the set of reals to the set of reals is not invertible, but if the codomain is restricted to the set of positive real numbers, the resulting function is invertible. As far a... | Invertible means one-to-one and onto. In particular, we only say that a map $f:A \to B$ is invertible if there is another map $g:B \to A$ such that both $f \circ g$ and $g \circ f$ are the identity maps over their respective spaces.
Of course, any one-to-one map can be made invertible by restricting the codomain to th... | {
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Transversals that are closed under multiplication in a group Let $H \le G$ be a group with subgroup $H$. A right (or left) transversal is a set of element which contains exactly one element from each right (or left) coset. Now for example for $S_3$ and $H = \{ (), (1 ~ 2) \}$ we have
$$
H, \quad H\cdot (1 ~ 2 ~ 3) = \... | If you don't require your group to be finite, then it is easy to find a counter-example : if $G=\mathbb{Z}$, then any subgroup $H$ will be of the form $n\mathbb{Z}$. Then if $A\subset G$ is transversal, it must have $n$ elements exactly, and thus it cannot be closed under addition since it is finite (and non reduced to... | {
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A sketch of proof of Prime Number Theorem I'm looking for some sketch of the elementary proof of the Prime Number Theorem, which would suffice to explain someone the general mechanism of proving PNT without going into deep analytic methods etc.
| Don Zagier has an article "Newman's Short Proof of the Prime Number Theorem", available for download here. It is an exposition of D.J.Newman's work, and consists of a self-contained four-page proof of the Prime Number Theorem. From the introduction:
We describe the resulting proof, which has a
beautifully simple str... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why $E_1=E\cup\bigcup_{i=1}^\infty G_i$? Let $E_k\supset E_{k+1}$ and $E=\bigcap_{i=1}^\infty E_i$. We set $$G_k=E_k\setminus E_{k+1}.$$
I don't understand why $$E_1=E\cup\bigcup_{i=1}^\infty G_i.$$
To me we simply have $E_1=\bigcup_{i=1}^\infty G_i$.
| Suppose that $x\in E$. Then for each $k\in\Bbb Z^+$ we have $x\in E_{k+1}$, and therefore $x\notin E_k\setminus E_{k+1}$, i.e., $x\notin G_k$. In other words, $x$ is not in any of the sets $G_k$, so $x\notin\bigcup_{k\ge 1}G_k$. On the other hand $x\in E_1$. Thus, $x\in E_1\setminus\bigcup_{k\ge 1}G_k$. The same is tru... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Topologically equivalent metrics? Ceiling function of metric $d$ I am asked if the following metrics are topologically equivalent or not.
$(X,d)$ is a metric space and $d$ is the metric. Define $\lceil{d} \rceil (x,y)$ := $\lceil{d(x,y)} \rceil$:$X \times X \rightarrow [0, \infty)$. Are $d$ and $\lceil{d} \rceil$ topo... | Hint: Consider the interval $[0,1]$ with the usual metric. What is the topology under the ceiling metric?
| {
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"timestamp": "2023-03-29T00:00:00",
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What is $0 \times \infty$? My question is - I know, $0\times anything=0$ and $anything \times \infty=\infty$.
So,what is $0 \times \infty$?
I suppose it's $0$ but why not $\infty$?
If I say that area of an indefinitely long line is $\infty*0=0$,where am I wrong?
I know upto limits and basic derivatives.
Thanks.
| $\infty$ is not a normal number and the rules of arithmetics only apply to normal numbers. The expression $0\times \infty$ is therefore not a sensible arithmetic expression that we can evaluate.
Instead, it is often used as a memonic when considering limits. If we have a sequence which grows without bounds (like $a_n=\... | {
"language": "en",
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Continuity of $L_p$ norm in $p$ with $\varepsilon$-$\delta$ definition Assume that $\|f\|_p< \infty$ for $1\le p<\infty$.
In this question we showed that
$$
g(p)=\|f\|_p
$$
is continuous in $p \ge 1$. The technique was to use Dominant Convergence theorem.
Using $\varepsilon$-$\delta$ language, what this means is that f... | Here's a super-soft answer. Fix a measurable function $f$ such that $f\in L^p$ for all $p\in (p_-, p_+)$ ($p_+$ possibly being $\infty$). Let
$$\Phi\left(\frac 1 p\right)=\left[ \int \lvert f\rvert^p\right]^\frac{1}{p}.$$
This function $\Phi$ is log-convex on the interval $\left(\frac1{p_+}, \frac1{p_-}\right)$, meani... | {
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Proof of a Basic Inequality I am new to this stack exchange and if I have any wrongdoing please let me know. My question is how to prove the following inequality:
$2^{n+1}>n^2$ assuming $n \in \mathbb{N}$
My thought is to prove this by mathematical induction.
Let $P(n)$ be the proposition
$P(1)$ is true as $4 = 2^2 > ... | Hint
$$2^{k+2}=2\cdot 2^{k+1}>2 \cdot k^2 \ge (k+1)^2$$ for $k\ge 3.$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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The angle between two rays in 3D space This is a problem from Mathematics GRE Subject Test - #42.
In the xyz-space, what is the degree measure of the angle between the rays $z= (x>=0), y=0$ and $z= (y>=0), x=0$?
a)0; b)30; c)45; d)60; e)90
My Attempt at a Solution
Because the first set of rays are always along the l... | Well, this might not be kosher but:
o = (0, 0, 0) is the vertex of the two rays. Let a = (1,0,1) is in Ray 1. Let b = (0, 1,1) by in ray two. The distance between a and o is $\sqrt{2}$. Between b and o is $\sqrt{2}$ and between a and b is $\sqrt{2}$. So the three points form an equilateral triangle. So the angle i... | {
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"url": "https://math.stackexchange.com/questions/1466204",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Can we predict the number of non-zero singular values in this case? If there are two matrices $P$ (dimensioned $m\times 1$) and $Q$ ($n\times1$) and a matrix $M$ is constructed by $M=PQ'$ (where the ' indicates transpose), so $M$ is of size $m\times n$.
Does $M$ have only one non zero singular value? If so, why?
All ... | You have, for any $X$ of size $m\times 1$,
$$
MM'X=PQ'QP'X=(Q'Q)(P'X)P
$$
(Note that $Q'Q$ and $P'X$ are $1\times1$, i.e. a scalar).
So if $Y$ is any eigenvector of $MM'$ with nonzero eigenvalue, i.e. $MM'Y=\lambda Y$, necessarily $Y$ is colinear with $P$, since we get $\lambda Y=(Q'Q)(P'Y)P$. Thus, $MM'$ can have a s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1466329",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Prove a function $f: \mathbb{R}^{2} \to \mathbb{R}^{2}$ is surjective I'm given the map $f: (x,y) \mapsto (x+3,4-y): \mathbb{R}^{2} \to \mathbb{R}^{2}$; how do I prove this function is onto (surjective)?
So far I said that let $x=z$ and $y=k$,
therefore $f(x,y)=(z+3,4-k)$, that means $f(x,y)$ is onto (surjective).
I'm ... | Let $(z_{1},z_{2}) \in \mathbb{R}^{2}$; then $z_{1} = x+3$ and $z_{2} = 4-y$ for some $(x,y) \in \mathbb{R}^{2}$ iff $x = z_{1}-3$ and $y = 4-z_{2}$; this shows that for every point $(z_{1},z_{2})$ of $\mathbb{R}^{2}$ there is some unique $(x,y) \in \mathbb{R}^{2}$ such that $(z_{1},z_{2}) = f(x,y)$, so $f$ is in fact ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1466433",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Subset of a finite group Let G be a finite group. Let $$H = \{b \in G.\ bab^{−1} \in \langle a \rangle \}.a \in G$$
Prove that if G is a finite group, then H is a subgroup of G.I think that a good approach is to prove that $$ba^i b^{-1} = (bab^{−1})^i \text{ for i an integer}$$
But I need a pointer on how to do that
| Try this:
$H $ is not empty as $e_G \in H$
Let $b_1,b_2\in H$ then $b_1ab_1^{-1},b_2ab_2^{-1}\in \langle a \rangle $. So let $b_1ab_1^{-1}=a^m$, $b_2ab_2^{-1}=a^p$
Now $b_1b_2a(b_1b_2)^{-1}=b_1(b_2ab_2^{-1})b_1^{-1}=b_1a^m b_1^{-1}=(b_1ab_2^{-1})^m=a^{pm}\in \langle a\rangle$. Thus $b_1b_2\in H$.
Again $a=b^{-1}a^mb$.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1466526",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Show that balls in $L^{1 + \delta}(\mu)$, with $\mu$ some finite measure, are uniformly integrable Can anyone give some suggestion/guideline to do this problem :
Suppose $\mu$ is a finite measure and for some $\delta > 0$ $$\sup_n \int |f_n|^{1 + \delta}d\mu < \infty.$$
Show that $\{f_n\}$ is uniformly integrable.
The ... | Let $H=\sup_{n} \int |f_{n}|^{1+\delta} $ and $E=E_{n,M}=\{ x:|f_{n}(x)|>M \}$.Then we have
$$M^{1+\delta} \mu(E) \leq \int_{E} |f_{n}|^{1+\delta} \leq H$$
and
$$\int_{E} |f_{n}| \leq \left( \int_{E} |f_{n}|^{1+\delta} \right)^{\frac{1}{1+\delta}} \left( \int_{E} 1 \right)^{\frac{\delta}{1+\delta}} \leq H^{\frac{1}{1+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1466589",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Let $G$ be a group and suppose that $a*b*c=e$ for all $a,b,c \in G$, show that $c*b*a=e$ I'm really in the dark here:
$a*b*c=e=identity$
$a*e=e*a, b*e=e*b, c*e=e*c$
$a*b*c=e$
$e=e$
$c*e*b=e*c*b$
$c*a*b*c*b=a*b*c*b*c*e=c*b$
$c*a*b*c*b=a*b*c*b*c*e=c*b*a*b*c$
$c*a*b*c*b=c*b*a*b*c$
$c*a*b*c*b*a*b*c=c*b*a*b*c$
$c*b*a*b=c*b*... | This is not true in general. Let $G$ be $S_3$, and take $a = (1,2)$, $b= (2, 3)$ and $c =(a*b)^{-1}= (1, 2, 3)^{-1}= (1, 3, 2)$. Then $a*b*c = e$, but $c$ is not the inverse of $b*a = (1, 2,3 )$, so $c*b*a \neq e$.
For a general example, let $a$ and $b$ be any two non-commuting elements in a group $G$, and let $c = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1466720",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
If $n$ divides $m$, then $n$ divides $m^2$ I have been asked in one of my problem sheets to prove that if $3$ divides $n$, then $3$ divides $n^2$.
So what I did was: Write $n=3d$, $d$ is an integer. So $n^2=9d^2$, therefore $n^2=3*3d^2=3c$, where $c$ is an integer.
QED.
But then the same method works in general for ... | You have no mistake,
$n^2 = n*n$
therefore, if m divides n:
$n=mc,c\in Z$
$n^2=m(mc^2)$
$mc^2\in Z$
therefore m divides $n^2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1466848",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Find the cost, given that reducing the selling price from 1080 to 1026 increased the loss by 4%
When a shopkeeper reduces the selling price of an article from 1080 to
1026 his loss increases by 4% . What is the cost price of article?
Solution of book :
4% CP = 1080-1026
CP = 1350
Easy enough. Now see very similar... | If I have understood correctly, the first method is not correct.
the word "loss" ought to refer to the difference between the cost price ($C$) and the sell price ($S$). For clarity, let's define $S$ to be the initial sell price, prior to any discounts. (So for the first problem $S=1080$ and for the second $S=1250$) Th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1466945",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Maximum area of a polygon inscribed in a complex triangle? Let $T$ be an acute triangle. Inscribe a pair $R,S$ of rectangles in $T$ as shown :
Let $A(x)$ denote area of polygon $X$ find the maximum value (or show that no maximum exists), of $\frac{A(R)+A(S)}{A(T)}$ where $T$ ranges over all triangles and $R,S$ overal... | Consider the sides as below shown below
As in the figure
$\frac{A(S)+A(R)}{A(T)} = \frac{ay+bz}{\frac{hx}{2}}$
Where $h=a+b+c$ the altitude of $T$.
By similar triangles we have,
$\frac{x}{h}=\frac{y}{b+c}=\frac{z}{c}$
So
$\frac{A(S)+A(R)}{A(T)} =\frac{2}{h^{2}} (ab+ac+ bc)$
we need to maximise $(ab+bc+ca)$ subject ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1467049",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find $M_{min}$ if there exist constant $M$ such $f(x)If $\dfrac{f(x)}{x^2}$monotone increasing function on $x\in (0,+\infty)$,and there exist constant $M$,such $f(x)<M,\forall x\in (0,+\infty)$,then Find the $M_{min}$
If we let $g(x)=\dfrac{f(x)}{x^2}$,then for any $x,y>0(x<y)$,we have $g(x)<g(y)$ or
$$\dfrac{f(x)}{x... | The function $f$ must not have an infimum. Take $f(x) := -\frac{1}{x}$, then $f(x)$ is bounded from above by $0$ and $\frac{f(x)}{x^2} = -\frac{1}{x^3}$ is monotonically increasing, but $f(x) \to - \infty$ as $x \to 0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1467166",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Finding condition for integral roots of a quadratic equation. I need to find the values of k(possible) for which the quadratic equation $$x^2+2kx+k =0$$ will have integral roots.
So I assumed roots to be $a,b$
Then I got the condition $a+b=-2k$and $a\cdot b=k$; so combining these I get $a+b+2ab=0$;
And now I need to fi... | Here is a hint to develop your existing method. Multiply the equation in $a$ and $b$ by $2$ and add a constant which enables you to factorise it.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1467285",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
About a set that is continuous image of a measurable set This is my first post. I hope that you can help me with a little hint. My problem says: If $S\subseteq \mathbb{R}$, $S^2$ is defined to be $S^2=\{s^2\ |\ s\in S \}$.Show that if $\lambda(S)=0$, then $\lambda(S^2)=0$, where $\lambda$ is Lebesgue measure.
I can pr... | Hint: Actually if $f\in C^1(\mathbb R),$ then $m(S)=0\implies m(f(S))=0$ and the general result is no harder to prove. For the proof, WLOG $S\subset [-a,a]$ for some $a>0.$ Use the boundedness of $f'$ on $[-a,a]$ to show there is $C$ such that $m(f(I)) \le Cm(I)$ for each interval $I\subset [-a,a].$ Therefore ...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1467391",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Is "polynomials in $x$" a monad? The construction of polynomials $R \mapsto R[x]$ gives a functor $P: \mathbf{Ring} \to \mathbf{Ring}$ on the category of possibly noncommutative rings. Choosing a ring $R$ for the moment, there is a nice homomorphism $R \to P(R)$ which embeds in the obvious way, taking $r$ to the const... | Yes, this is a monad. Much more generally, if $\mathcal{C}$ is a monoidal category and $M$ is a monoid object in $\mathcal{C}$, then the functor $P(R)=M\otimes R$ is a monad using the monoid structure of $M$. In this case, $\mathcal{C}=\mathbf{Ring}$, the monoidal structure is $\otimes_\mathbb{Z}$, and $M=\mathbb{Z}[... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1467500",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 1,
"answer_id": 0
} |
How to integrate $\int(x\pi-x^2)\cos(kx)dx$ My question is can I solve by integrating by parts if I do $u = (x\pi-x^2)$,or first I have to do $∫[x\pi \cos(kx)-x^2\cos(kx)]dx$ and then split it in two integrals $\int x\pi\cos(kx)dx-\int x^2\cos(kx)dx$ ?
| You may directly integrate by parts twice,
$$
\int(\pi x-x^2) \cos (kx)\:dx=\frac1k(\pi x-x^2)\sin (k x)-\frac1k\int(\pi-2x) \sin (kx)\:dx
$$ then
$$
\begin{align}
&\int(\pi x-x^2) \cos (kx)\:dx\\&=\frac1k(\pi x-x^2)\sin (k x)-\frac1k\left((\pi-2x)(-\frac1k \cos (kx))+\int(-2) \frac1k \cos (kx)\:dx\right)
\end{align}
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1467605",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Why is $f(x) \delta(x) = f(0)\delta(x)$ only true when $x=0$? This is a follow up from a previous question asked by me.
I know that $$\delta(x) = \begin{cases} 0 & \space \mathrm{for} \space x \ne 0 \\\infty&\ \mathrm{for} \space x = 0 \end{cases} $$ and that
$$\int_{-\infty}^{\infty} \delta(x) \mathrm{d}x = 1$$
I a... |
PRIMER:
In This Answer and This Answer, I provided more detailed primers on the Dirac Delta. Herein, we condense the content of those answers.
The Dirac is not a function, but rather a Generalized Function also known as a Distribution.
The symbol $\int_{-\infty}^{\infty}\delta (x)f(x)\,dx$ is ,in fact, not an integ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1467719",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Limit: $\lim_{x\to 0}\frac{\tan3x}{\sin2x}$ $\lim_{x\to 0}\frac{\tan3x}{\sin2x}$= $\lim_{x\to 0}\frac{\frac{\sin(3x)}{\cos(3x)}}{\sin2x}=\lim_{x\to 0}\frac{\sin3x}{1}\cdot\frac{1}{\cos(3x)}\cdot\frac{1}{\sin(2x)}$
From this point I am lost. I believe I can pull a 3 and 2 out but I am not sure how. Can someone give me d... | $\lim_{x\to 0}\frac{\tan3x}{\sin2x}=\lim_{x\to 0}\frac{\tan3x}{3x}\frac{2x}{\sin2x}\frac{3x}{2x}=\frac{3}{2}$
or by using L'Hôpital's rule
$\lim_{x\to 0}\frac{\tan3x}{\sin2x}=\lim_{x\to 0}\frac{3(1+\tan^2 3x)}{2\cos2x}=\frac{3(1+\tan^2 3(0))}{2\cos2(0)}=\frac{3}{2}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1467848",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Combinations confusion for coin flips 10 fair coins are tossed. How many outcomes have 3 Heads?
I'm supposed to solve it with combination C(10, 3). But...
How do you know it's a combination that will solve it? I'm not interested in what makes it a combination, instead of a permutation. I know the answer is (some #)/3^1... | think about naming the order of the tosses ... toss#1, toss#2 etc.
e.g. The number of ways of getting 3 heads when tossing 5 coins is the same as the number of ways of deciding which 3 of the 5 tosses came up heads
e.g. the choice ${2,4,5}$ corresponds to the sequence THTHH
the choice ${1,2,5}$ corresponds to the sequ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1467946",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Determinant of transpose intuitive proof We are using Artin's Algebra book for our Linear Algebra course. In Artin, det(A^T) = det(A) is proved using elementary matrices and invertibility. All of us feel that there should be a 'deeper' or a more fundamental or a more intuitive proof without using elementary matrices or... | The determinant of a matrix does not change when you compute it via cofactor expansion along column or row. Thus expanding along a row in $A$ is equivalent to expanding along a column in $A^t$. I'm not sure if this is what you meant by "using invertibility".
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1468064",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
How many solutions does the equation x + y + w + z = 15 have if x, y, w, z are all non-negative integers? Combinatorics question:
What I tried for solving this problem is (16 - 1 + 4 choose 4). I got 16 from the numbers 0 thought 16 as possible values for x, y , w or z.
However apparently the answer is (16 - 1 + 3 cho... | Think of $15$ as a sequence of stars. You can insert $3$ bars in any position between them to get a solution, for example $0+3+10+2$ would be represented this way:
$$|\star\star\star|\star\star\star\star\star\star\star\star\star\star|\star\star$$
It should be clear than any permutation of those stars and bars (which is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1468143",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Is there a way to show that $\gcd(a,b) = ax + by $ without also showing that its the smallest positive linear combination? Is there a way to show that $\gcd(a,b) = ax + by$ without also showing that it is the smallest positive linear combination? i.e. Can it be shown that there exists an $a$ and $b$ such that $\gcd(a,b... | Let's revise the definition of GCD.
Let $a, b$ be integers with at least one of them being non-zero. A positive integer $d$ is said to be the GCD of $a$ and $b$ and denoted by $d = (a, b)$
*
*if $d \mid a, d \mid b$
*if there is any integer $c$ with $c \mid a, c \mid b$ then $d \mid c$
Let $b$ be the non-zero inte... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1468283",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
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