Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Integration and evaluation I have to evaluate this integral here
$$\int_x^\infty \frac{1}{1+e^t} dt$$
I've tried with the substitution $e^t=k$, but at the end I get $\log|k|-\log|1+k| $, which evaluated between the extrema doesn't converge... I hope that somebody can help!
| Hint:
$$\frac { 1 }{ 1+{ e }^{ t } } =\frac { 1 }{ { e }^{ t }\left( 1+{ e }^{ -t } \right) } =\frac { { e }^{ -t } }{ 1+{ e }^{ -t } } \\ \int { \frac { dt }{ 1+{ e }^{ t } } } =-\int { \frac { d\left( 1+{ e }^{ -t } \right) }{ 1+{ e }^{ -t } } } $$
| {
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"timestamp": "2023-03-29T00:00:00",
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Rewrite a function of $x+iy$ in terms of $z$ only How can you rewrite a function like
$$f(x+iy)=e^{2xy}\cos(x^2-y^2)-i\left[e^{2xy}\sin(x^2-y^2)+C\right]$$
in terms of just $z$? Are there any tricks that you can use or is it just the fact that you need to recognize the functions when you see them?
| Suppose we have an expression for $f$ in terms of $z$. If we restrict $z$ to be real, then we are effectively replacing the complex variable $z$ by the expression $x + 0i$ where $x$ is a real variable. What we end up with is an expression for $f$ along the real line in terms of $x$. What is miraculous is that we can r... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Given that $f:G\to G$ on a group $G$, defined by $f(x)=x^3$, is an isomorphism, how do I show that $G$ is abelian? Given that $f:G\to G$ on a group $G$, defined by $f(x)=x^3$, is an isomorphism, how do I show that $G$ is abelian, please?
| Answer of the changed question: Note that for any $a,b \in G$, $xy=f(x^{-1})f(y^{-1})=f(x^{-1}y^{-1})=f((yx)^{-1})=yx$.Hence $G$ is abelian.
| {
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Prove that $f$ is continuous Let $f$ be a non-constant function such that $f(x+y)=f(x)\cdot f(y),\forall x,y\in\mathbb{R}$ with $\lim_\limits{x\to 0}{f(x)}=l\in\mathbb{R}$. Prove that:
a) $\lim_\limits{x\to x_0}{f(x)}\in\mathbb{R},\forall x_0\in\mathbb{R}$
b) $l=1$
I have managed to prove a:
$$\lim_\limits{h\to 0}{f(x_... | Assuming that you have established part a) and the fact that $f(0) = 1$ we can proceed to show that $l = 1$. Clearly we can see that $f(x)f(x) = f(2x)$ and hence taking limits when $x \to 0$ (so that $2x \to 0$) we get $l^{2} = l$. So either $l = 0$ or $l = 1$.
Again note that $f(x)f(-x) = f(0) = 1$ and taking limits a... | {
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Prove that $g$ is continuous at $x=0$ Given, $g(x) = \frac{1}{1-x} + 1$. I want to prove that $g$ is continuous at $x=0$. I specifically want to do an $\epsilon-\delta$ proof. Related to this Is $g(x)\equiv f(x,1) = \frac{1}{1-x}+1$ increasing or decreasing? differentiable $x=1$?.
My work: Let $\epsilon >0$ be given.... | Hint. See my comment to your post.
After doing your algebra, as you have shown,
$$\left|\dfrac{1}{1-x}+1-2\right| = \left|x\right|\left|\dfrac{1}{1-x}\right|\text{.}$$
We have $|x| < \delta$. Let $\delta = 1/2$, then
$$|x| < \delta \implies -\delta < x < \delta \implies 1-\delta < 1-x < 1 + \delta \implies \dfrac{1}{1... | {
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probability that exactly two envelopes will contain a card with a matching color Suppose that 10 cards, of which 5 are red and 5 are green, are placed at random in 10 envelopes, of which 5 are red and 5 are green. Determine the probability that exactly two envelopes will contain a card with a matching color.
clearly $\... | Suppose both the cards and envelopes are rearranged (without changing the pairings) so that all red envelopes are on the left and all green envelopes are on the left. Now focusing on the cards on the left, we want the probability that exactly one of these is red (then by symmetry there will also be one matching pair on... | {
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"question_score": "4",
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Show that $H+K$ is a subspace of $V$ I am trying to solve this, please give me hints.
$H,K \in V$, $V$ is a vector space.
$H+K=\{w:w=u+v:u \in H, v \in K \}$
Show that $H+K$ is a subspace of $V$.
| A sufficient condition is that $H$ and $K$ are subspaces of $V$. It suffices to check if we have $aw_{1} + bw_{2} \in H+K$ for all scalars $a,b$ and all $w_{1},w_{2} \in H+K$.
Let $w_{1},w_{2} \in H+K$ and let $a,b$ be scalars. Then $w_{1} = h_{1} + k_{1}$, $w_{2} = h_{2}+k_{2}$ for some $h_{1},h_{2} \in H$ and some $k... | {
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Algebra with complex numbers (i) My professor has assigned this problem:
Let r be a real number. By using r = r + 0i, confirm that r(a + bi) = (ra) + (rb)i.
My understanding is that this looks at distributive property, but I do not understand how r = r + 0i relates. Could someone provide a jumping off point? I don't... | I guess the professor defined "complex-multiplication" operation using "real-multiplication" as follows:
$(a+ib) \cdot (c+ id):= (ac-bd) + i(ad+bc)$.
Hence, $r\cdot (a+ib) = (r+i0)\cdot (a+ib)= (ra -0b) + i(rb + 0a)= (ra)+ i(rb)$.
| {
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Bounds for the PDF of $(X,XY)$ if the PDF of $(X,Y)$ has support $0\leq X\leq Y\leq1$
Given the PDF $\,f_{XY}(x,y)=8xy,\, 0\leq x \leq y \leq 1$. Let $U=X$ and $V=XY$. Find the pdf of $(U,V)$.
My answer:
So, $X=U$ and $Y= V/U$, The Jacobian J of the inverse transformation would then equal:
$J=\frac{1}{u}$..., then
H... | $U=X$ so the range for $U$ is $0\lt U\lt 1$.
$V=XY$ has its maximum when $Y=1$, giving $V=X=U$. $V$ has its minimum when $Y=X$, giving $V=X^2=U^2$.
So the joint pdf for $U,V$ is
$$f_{U,V}(u,v) = |J|f_{X,Y}(x(u,v),\;y(u,v)) = \dfrac{8v}{u},\quad 0\lt u\lt 1\; \text{ and } u^2\lt v\lt u.$$
| {
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"timestamp": "2023-03-29T00:00:00",
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prove the combinatorial identity: $ {2n \choose n} + {2n\choose n-1} = (1/2){2n + 2 \choose n + 1}$ So the first thing I did was to multiply and then isolate the left-most term.
That leaves me with $$ 2{2n \choose n} ={2n + 2\choose n + 1} - 2{2n \choose n -1} $$
No specific reasoning except that it looks cleaner, at... | Whenever you see these types of identities, at least I don't think like that. I think algebraically. Maybe long, but always correct. We want to prove
$$2{2n \choose n} ={2n + 2\choose n + 1} - 2{2n \choose n -1}$$
We know that $${2n\choose n}+{2n\choose n-1}={2n+1\choose n}$$
Multiplying by $2$ $$2{2n\choose n}+2{2n\ch... | {
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The order of sum of powers? For example, the sum of n is n(n+1)/2, the dominating term is n square(let say this is order 2). For the sum of n^2, the order is 3. Then for the sum of n^k, is the order k+1?
I been searching Faulhaber's formula and Bernoulli numbers, I'm not sure what is the order of it. It's much more co... | By the definition of Riemann integral, $\frac1n \sum_{k=1}^n (\frac{k}{n})^\alpha \rightarrow \int_0^1 t^\alpha dt=\frac1{\alpha+1}$.
Then $\sum_{k=1}^n k^\alpha \sim \frac{n^{\alpha+1}}{\alpha+1}$
| {
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$G$ solvable $\implies$ composition factors of $G$ are of prime order. I'm trying to prove the equivalency of the following definitions for a finite group, $G$:
(i) $G$ is solvable, i.e. there exists a chain of subgroups $1 = G_0 \trianglelefteq G_1 \trianglelefteq \cdots \trianglelefteq G_s = G $ such that $G_{i+1}/ ... | Hint: Use induction on the order of $G$:If $G_{s-1}$ is a proper normal subgroup of $G$ such that $G/G_{s-1}$ is abelian, then there are two cases:
*
*Either $1 < G_{s-1}$; then by induction, the composition factors of $G_{s-1}$ and $G/G_{s-1}$ have prime orders.
*Or $1 = G_{s-1}$ in which case $G$ is itself abelia... | {
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Show set is countable I want to show that the set
$A=\{n^2+m^2:n,m\in \mathbb{N}\}$
is countable.
Is it enough to state that, since $\mathbb{N}$ is closed under multiplication and addition, the set A must be a subset of $\mathbb{N}$; and since any subset of a countable set is countable, A must be countable?
| Yes. Unless you define "countable" as "infinite and countable", in which case you need to prove that $A$ is infinite (which shoud be almost trivial).
| {
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Proving that $\pi(2n) < \pi(n)+\frac{2n}{\log_2(n)}$ Given that $\pi(x)$ is the prime-counting function, prove that, for $n\geq 3$,
1: $\pi(2n) < \pi(n)+\frac{2n}{\log_2(n)}$
2:$ \pi(2^n) < \frac{2^{n+1}\log_2(n-1)}{n}$
For $x\geq8$ a real number, prove that
$\pi(x) < \frac{2x\log_2(\log_2(x)}{\log_2(x)}$.
I understand... | We note that$$\binom{2n}{n} \le 4^n$$and is divisible by all primes such that $n < p < 2n$.
Prove that if $n \ge 3$ is an integer, then$$\pi(2n) < \pi(n) + {{2n}\over{\log_2(n)}}.$$
$\pi(2n) - \pi(n)$ is the number of primes such that $n < p < 2n$. If there are $k$ such primes, we have the inequality$$n^k \le 4^n \im... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1434928",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "4",
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Understanding one point compactification I just read about one point compactification and i am having some difficulty in grasping the concept.
Does one point compactification mean that we are simply adding a point to a non compact space to make it compact.
For example, my book says that $S^n$ is the one point compactif... | No, you don't just add a point. You have to define the topology in the newly created space (set). In case of $\mathbb{R}^n $ one thinks of this as adding a point at $\infty$. A topology may be defined by saying that set is, by definition, a neighbourhood of this point if it contains the exterior of some closed ball $B_... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Evaluate the following definite integrals using the Fundamental Theorem of Calculus Evaluate the following definite integrals using the Fundamental Theorem of Calculus
$$ \int_{-10}^1 s | 25 - s^2 | \; \mathrm d s. $$
my work:
$$ s=\pm 5 $$
$$ \int^{-5}_{-10} f(s) + \int^5_{-5} f(s) + \int^1_5 f(s) $$
Stuck here. Can't... | Hint: See what is the sign of $25-s^2$ in each of the intervals for $s$ and write down the module $|25-s^2|$
| {
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Show the Set is Connected Let $f: \mathbb{R} \to \mathbb{R}$ be continuous. Show the set
$$A:= \{\alpha \in \mathbb{R}: \exists \{x_n\}_{n \in \mathbb{N}} \subset \mathbb{R} \, \, \text{ with} \, \, \lim_{n \to \infty} f(x_n) = \alpha \}$$
is connected.
Proof idea:
Since this is a subset of $\mathbb{R}$. This is... | $A$ is the closer of $f(\Bbb R)$ i.e $A=\overline{f(\Bbb R)}$.
As $\Bbb R$ is connected and $f$ continuous, then $f(\Bbb R)$ is connected, hence the closer is connected.
| {
"language": "en",
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Gaussian elimination involving parameters The problem is :Solve the given system of equations involving the parameter a :
$$x+y+az=1\\
x+ay+z=a\\
ax+y+z=a^2\\
ax+ay+az=a^3 .$$
I tried to solve this using the Gaussian method but I'm stuck because this is $4\times3$ matrix, and the Gaussian process is used for square mat... | Subsitute $y=1-x-az$ from the first equation and then $z= - (a - x + 1)$ from the
second equation, assuming $a\neq 1$. Then the third equation gives $x=(a^2+2a+1)/(a+2)$, the case $a=-2$ being impossible. Then the fourth equation gives
$$
a(a+1)(a-1)=0.
$$
For $a=0$ we have a $3\times 3$ system with unique solution $(... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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If X and Y are equal almost surely, then they have the same distribution According to this question,
If $2$ r.v are equal a.s. can we write $\mathbb P((X\in B)\triangle (Y\in B))=0$
why is this so?
I get the equivalent statement $\mathbb P([(X\in B)\triangle (Y\in B)]^C)=1$ intuitively, but I don't see how to show ri... | $$\{X=Y\} \subseteq [(X\in B)\triangle (Y\in B)]^C$$
Suppose $\omega \in \{X=Y\}$. To show that $\omega \in [(X\in B)\triangle (Y\in B)]^C$.
We have to show that $\omega$ belongs to both of the following sets
$X \in B \cup Y \notin B$
$X \notin B \cup Y \in B$
For $X \in B \cup Y \notin B$, we have two cases by $X$: $... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Prove or disprove that $K$ is a subgroup of $G$.
Let $H$ be a subgroup of $G$, let $a$ be a fixed element of $G$, and let $K$ be the set of all elements of the form $aha^{-1}$, where $h \in H$. That is $$K = \{x \in G~ : x=aha^{-1}~ \text{for some }h\in H \}$$ Prove or disprove that $K$ is a subgroup of $G$.
I have a... | Consider the mapping $f\colon G\to G$ defined by
$$
f(x)=axa^{-1}
$$
Then, for $x,y\in G$,
$$
f(x)f(y)=(axa^{-1})(aya^{-1})=a(xy)a^{-1}=f(xy)
$$
Thus $f$ is a homomorphism and $K=f(H)$, so $K$ is a subgroup.
Actually, since $f$ is even an automorphism, you can say that $K=f(H)$ is a subgroup if and only if $H$ is a sub... | {
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Iterating limit points ad infinitum to obtain $\{0\}$ Let $X\subseteq\mathbb{R}$. We define $D(X):=\left\{x\mid x\in \mathbb{R}\mid \forall n\in\mathbb{N}\;\exists y\in X\left[0<|x-y|<\frac1{2^n}\right]\right\}$, i.e. the set of limit points of $X$.
We can iterate $D$ by: $D^{(2)}(X):= D(D(X)), \ldots$. We can then al... | We can solve this by looking at your $X_1 = \{ \frac{1}{2^n} | n \in \mathbb{N} \}$. For every element $\frac{1}{2^n}$ in $X_1$ we can find some $\epsilon_n$ such that $[\frac{1}{2^n}, \frac{1}{2^n} + \epsilon_n] \cap X_1 = \frac{1}{2^n}$. We can then map $[0,1] \to [\frac{1}{2^n}, \frac{1}{2^n} + \epsilon_n]$ bijectiv... | {
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Evaluate $\int \tan^6x\sec^3x \ \mathrm{d}x$ Integrate $$\int \tan^6x\sec^3x \ \mathrm{d}x$$
I tried to split integral to $$\tan^6x\sec^2x\sec x$$ but no luck for me. Help thanks
| Here is what I got from Wolfram Alpha, condensed a little bit
$$\int tan^6(x)sec^3(x)dx$$
$$= \int (sec^2-1)^3sec^3(x)dx$$
$$= \int \Big(sec^9(x) -3sec^7(x)+3sec^5(x)-sec^3(x)\Big)$$
$$= \int sec^9(x)dx -3\int sec^7(x)dx+3\int sec^5(x)dx-\int sec^3(x)dx$$
Since $\int sec^m(x) = \frac{sin(x)sec^{m-1}(x)}{m-1} + \frac{m-... | {
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about necessary and sufficient condition, again Here I have a sentence picked up from a first year book:
The statement “if A then B” is equivalent to the statement “A is a sufficient condition for B” and to the statement “B is a necessary condition for A"
I understand the first part, but I cannot see how the statemen... | Here's my understanding:
It means that $A$ necessarily entails $B$.
The first statement tells you that it's enough to have $A$ to get $B$. But you could have $B$ without $A$.
The second statement tells you that given $A$, you have to have $B$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How do I prove the norm of this linear functional is $2$? $f$ is defined on $C[-1, 1]$ by $$f(x)=\int_{-1}^0 x(t) dt - \int_0^1 x(t) dt.$$
I can show that $\|f\| \le 2$. I don't know how to show $\|f\| \ge 2$.
| Of course if you can plug in
$$x(t) = \begin{cases} 1 & \text{if } t\in [-1, 0] \\ -1 & \text{if } t\in (0,1]\end{cases}$$
Then this function satifies $f(x) = 2$. But this $x$ is not continuous. However you can approximate this by $x_n \in C([-1,1])$, where
$$x_n (t) = \begin{cases}
1 & \text{if } t\in [-1, -\frac 1n)... | {
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Circular Banked Track Friction
I've tried answering this question by resolving forces, then finding an expression for friction and inserting the given data so I can prove $F = 0$. However, I never get an answer of $0$.
How do I do it?
| Notice, mass of the car $m=1\ t=1000\ kg$,
speed of the car $v=72\ km/hr=20\ m/sec$,
radius of the circular path $r=160\ m$
$$\tan\alpha=\frac{1}{4}\iff \sin\alpha=\frac{1}{\sqrt {17}}, \cos \alpha=\frac{4}{\sqrt{17}}$$
$\color{red}{\text{Method 1}}$:Resolving components
Component of centrifugal force acting on t... | {
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"timestamp": "2023-03-29T00:00:00",
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Is null matrix skew-symmetric My question is:
Is null matrix skew-symmetric?
I think it is true as A'=-A is trivially satisfied for a null matrix.
Please help.
| To test if a matrix is skew symmetric, you need to show that:
$$-A=A^T$$
For example in the case of $2\times 2$ matrices, you need:
$$\begin{bmatrix}-a&-b\\-c&-d\end{bmatrix}=\begin{bmatrix}a&c\\b&d\end{bmatrix}$$
When we set $a=b=c=d=0$ this is trivially satisfied.
Also this is satisfied non-trivially for example in ... | {
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"timestamp": "2023-03-29T00:00:00",
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Parameters such that the matrix is diagonizable Given the matrix
$$A= \begin{pmatrix}
1 & 0 & 0\\
k & 2 &0 \\
m& n & 1
\end{pmatrix}$$
where $k, m, n \in \mathbb{R}$ find the values of $k, m, n $ such that the matrix is diagonizable.
Solution
The eigenvalues of $A$ are the solutions of the equation:
$$\begin{vmatrix... | The system of equations that define the kernel of $A-I$ must have rank $1$ for $\ker A-I$ to have dimension 2, because of the rank-nullity theorem. This means the condition is the vectors $(k,1)$ and $(m,n)$ must be colinear. Explicitly:
$$kn=m.$$
You should be able to check that, given that relation, the minimal poly... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1436387",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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differential equation $y'=\sqrt{|y|(1-y)}$ Consider the differential equation
$y'=\sqrt{|y|(1-y)}$ with $y\le1$
I know that $y=0$ and $y=1$ are solutions.
The problem is to compute the general solution for $0<y<1$. On which interval is the solution defined?
I don't know where or how to start.
$y$ is always positive, s... | Given $$\displaystyle \frac{dy}{dx} = \sqrt{|y|\cdot (1-y)}=\sqrt{y(1-y)}\;,$$ bcz $\; 0<y<1$
So $$\displaystyle \frac{dy}{\sqrt{y(1-y)}} = dx\Rightarrow \int \frac{1}{\sqrt{y(1-y)}}dy = \int dx$$
Now Put $$\displaystyle y=\left(z+\frac{1}{2}\right)\;,$$ Then $dy=dz$
So we get $$\displaystyle \int\frac{1}{\sqrt{\left(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1436509",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Hash Collision Probability Approximation If an item is chosen at random $k$ times from a set of $n$ items, the probability the chosen items are all different is exactly $\dfrac{n^\underline{k}}{n^k}=\dfrac{n!}{(n-k)!n^k}$. For large $n$, the expression is said to be approximately equal to $\exp\left(\dfrac{-k(k-1)}{2n}... | The probability of choosing different items is
$$
\left(1-\frac0n\right)\left(1-\frac1n\right)\left(1-\frac2n\right)\cdots\left(1-\frac{k-1}n\right)\;.
$$
For $n\gg k^2$, we can keep only the terms up to first order in $\frac1n$ to obtain
$$
1-\frac1n\sum_{j=0}^{k-1}j=1-\frac{(k-1)k}{2n}\;.
$$
To get the exponential fo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1436603",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Measurability of the supremum of a Brownian motion After reading some text books about Brownian Motion i often encountered the following object
$$
\sup_{t \in [0, T]} B_t,
$$
where $(B_t)_{t \geq 0}$ is a Brownian Motion.
But how do i see that this object is measurable? The problem is that the supremum is taken over a... | Typically a Brownian motion is defined to have continuous sample paths. If you take some countable, dense subset $S\subset [0,T]$, you then have
$$\sup_{t\in S} B_t = \sup_{t\in [0,T]} B_t$$
by continuity of $B_t$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1436715",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
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For what values $m \in \mathbb{N}$, $\phi(m) | m$, where $\phi(m)$ is the Euler function. I am working with elementary number theory and, although in theory the $\phi$ Euler function seems easy to understood, I am having some problemas making the exercises.
For example, in this question:
For what values $m \in \mathbb... | $m=1$ works. Let $m=p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_k^{\alpha_k}$ with $2\le p_1<p_2<\cdots <p_k$. Then
$$\phi(m)=p_1^{\alpha_1-1}p_2^{\alpha_2-1}\cdots p_k^{\alpha_k-1}\left(p_1-1\right)\left(p_2-1\right)\cdots\left(p_k-1\right)$$
$$\phi(m)\mid m\iff \frac{p_1p_2\cdots p_k}{\left(p_1-1\right)\left(p_2-1\right)\c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1436803",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Support of a measure and set where it is concentrated I have some problems understanding the difference between the notion of support of a measure and saying that a measure is concentrated on a certain set.
If I have a $\sigma$-algebra and $\mu$ a measure, is it equivalent to ask the support of $\mu $ to be contained i... | Some supplementary information regarding Dominik's answer.
1) For most authors and in most cases of practical interest, the definition of support given by Dominik is appropriate. However, be aware that some authors do not distinguish between supports of a measure $\mu$ and sets where the measure $\mu$ concentrates --... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What is a permutation pattern? The wikipedia entry for permutation pattern gives this as an example:
For example, in the permutation π = 391867452, π1=3 and π9=2. A permutation π is said to contain the permutation σ if there exists a subsequence of (not necessarily consecutive) entries of π that has the same relative ... | Rather than thinking of a pattern as a single permutation, think of it as an equivalence class with respect to order isomorphism (i.e. if positions compare in the same way, then the entries in those positions compare in the same way, too). That equivalence class has a canonical representative where the $k$th smallest p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1437012",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show that the area under the curve of a measurabe function is Lebesgue measurable Problem: Let $E \subset R^n$ be a Lebesgue measurable set and $f : E \to [0,\infty)$ be a Lebesgue measurable function. Suppose
$$A = \{(x,y) \in R^{n+1} : 0 \le y \le f(x), x\in E\}.$$
Let $\lambda_1$ denote Lebsgue measure in $R^1$, $\l... | For $(a)$, notice that the result is obvious if $f$ is a simple function. Indeed the region under the graph of a non-negative simple function looks like the disjoint union of sets of the form $A_i \times [0,a_i]$, which are of course measurable. Now let $f$ be any non-negative measurable function and let $\{s_n\}$ be a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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How to factor$ x^2+2xy+y^2 $ I know the formula $x^2 + 2xy + y^2 = (x + y)^2$ by heart and in order to understand it better I am trying to solve the formula myself; however, I don't understand how step 4 is derived from step 3.
$$x^2+2xy+y^2\qquad1$$
$$x^2+xy+xy+y^2\qquad2$$
$$x(x+y) + y(x+y) \qquad3$$
$$(x+y)(x+y) \qq... | $$x\color{blue}{(x+y)} + y \color{blue}{(x+y)} \qquad3$$
Let $\color{blue}{b=(x+y)}$
So, now you have.
$$x\color{blue}{(b)} + y\color{blue}{(b)} = x\color{blue}{b} +y\color{blue}{b} =\color{blue}{b}x+\color{blue}{b}y $$
Now factor out the $\color{blue}{b}# :
$$\color{blue}{b} (x+y)$$
Remember, $\color{blue}{b=x+y}$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Velocity in applications of integration The question reads as this:
Person A starts riding a bike at noon (t=0) from Niwot to Berthoud, a distance of 20km, with velocity $v(t)=15/(t+1)^2$.
Person B starts riding a bike at noon as well (t=0) from Berthoud to Niwot with velocity $u(t)=20/(t+1)^2$
Assume distance is measu... | Let's say that Niwot is at (0,0) and Berthoud is at (20,0) giving the following:
$$A.)\quad P_v = \int_{0}^{t} V_{vt} = \int_{0}^{t}\frac{15\,dt}{(t+1)^2} = \frac{15t}{t+1}$$
$$B.) \quad P_u = 20 - \int_{0}^{t} V_{ut} = 20 -\int_{0}^{t}\frac{20\,dt}{(t+1)^2} = 20-\frac{20t}{t+1}$$
$$C.) \quad P_u = P_v\,\, \therefore ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Explicitly give an infinity of rational numbers who come from below the number $9.123412341234 ...$, Explicitly give an infinity of rational numbers who come from below the number $9.123412341234 ...$, and an infinity who come from above.
I know this number is $\frac{91225}{9999}$, but how can I give that set of number... | The set of rationals $\frac{n}{9999}$ where $n < 91225$ and $n$ is an integer is countably infinite with a cardinality of $ \aleph_0$. Likewise, the set of rationals $\frac{n}{9999}$ where $n > 91225$ and $n$ is an integer is countably infinite with a cardinality of $ \aleph_0$. In general, if you have a rational numbe... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$C(\textbf{R})$ is a vector subspace of $\textbf{R}^\textbf{R}$? How do I see that $C(\textbf{R})$ is a vector subspace of $\textbf{R}^\textbf{R}$?
| Let $\mathcal V$ be a vector space over a field $\mathbb F$. To check if a subset $\mathcal W\subset \mathcal V$ is a subspace of $\mathcal V$ it is sufficient to check:
*
*$0\in \mathcal W$
*$v,w\in\mathcal W\implies v+w\in\mathcal W$ for all $v,w\in\mathcal W$
*$v\in\mathcal W, a\in\mathbb F\implies a\cdot v\in\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How can the proof by induction be reliable when it depends on the number of steps? Yesterday, I got a math problem as follows.
Determine with proof whether $\tan 1^\circ$ is an irrational or a rational number?
My solution (method A)
I solved it with the following ways.
I can prove that $\tan 3^\circ$ is an irration... | Let $S_n$ be the statement that $\tan n^\circ$ is rational. Then
$$
\tan (n+1)^\circ = \frac{\tan n^\circ + \tan 1^\circ}{1-\tan n^\circ \tan 1^\circ}
$$
shows that $S_1,S_n\implies S_{n+1}$. So by assuming $S_1$ we have by induction.
$$
S_1\wedge S_1\implies S_1\wedge S_2\implies ...\implies S_1\wedge S_n\text{ etc.}
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1437679",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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If $z$ lies on the circle $|z-1|=1$ then$\frac {z-2} z$ is a purely imaginary number If $z$ lies on the circle $|z-1|=1$ then $\frac {z-2} z$ is a purely imaginary number. This is what by book states. Did'nt understand why. Can someone help?
Actually I was thinking of a more geometrical approach to the problem, as poin... | Let $z = x + iy$. Then $|z - 1|^2 = 1 \rightarrow (x-1)^2 + y^2 = 1$.
Then show that the real part of $\frac{z-2}{z}$ is $x(x - 2) + y^2$, which is zero, from above.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1437769",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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Confusing probability question I have got a task, which seems a quite confusing for me. It is simple: In a market, they sell eggs in egg holders, they store $10$ of them in each. There is $60$% chance, that all of the eggs are ok, $30$% chance, that exactly $1$ of them is broken, and $10$% chance, that exactly $2$ of t... | I see this as analogous to the pancake problem ... specifically, I would argue that having randomly observed a broken egg, the chances that you are in case C have disproportionately increased.
Imagine that you had $100$ of your cartons, $60$ of type A, $30$ of type B, and $10$ of type C. You select one egg randomly ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
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"answer_id": 1
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Does the inner product $\langle \cdot, \cdot \rangle$ induce any other norms other than the 2 norm? In the lecture my professor wrote that the standard inner product on $R^n$ is given by
$\langle x, y \rangle = x^Ty = \sum\limits_{i=1}^n x_i y_i$
which induces a norm $\sqrt{\langle x,x \rangle} = \|x\|_2$
My question... | This is a really interesting question, and here is a partial answer. The $1$ and $\infty$ norms do not come from inner products. For a norm to have an associated inner product actually gives you a lot of structure. For example (if the scalars are real for convenience),
$$\left\| x - y \right\|^2 = \langle x - y, x -... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1437995",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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How to find the maximum curvature of $y=e^x$? So I found the curvature to be $K = \dfrac{e^{x}}{(1+e^{2x})^{3/2}}$ but I don't know how to maximize this?
| We can differentiate and set equal to zero to get
$$\frac{e^x-2e^{3x}}{(1+e^{2x})^{5/2}}=0\implies e^x=2e^{3x}\implies\frac{1}{2}=e^{2x}.\tag{1}$$
Then solve to obtain $x=-\frac{1}{2}\log 2$.
Note - the various algebraic operations in (1) are possible since $e^x\neq 0$ for all $x\in\mathbb{R}$. For future similar quest... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1438119",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Are the perfect groups linearly primitive? A finite group $G$ is perfect if $G = G^{(1)} := \langle [G,G] \rangle$, or equivalently, if any $1$-dimensional complex representation is trivial.
A finite group $G$ is linearly primitive if it has a faithful complex irreducible representation.
Question: Are the perfect fin... | No. Any linearly primitive group must have cyclic center by Schur's lemma, but there's no reason a perfect group should have this property. I think that, for example, the universal central extension of $A_5 \times A_5$ is perfect but has center $C_2 \times C_2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1438207",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Limit of a sequence of integrals and the integral of a limit For every $n \in \mathbb{Z}_{>0}$ define
$$
f_n(x) = \frac{x^n}{1+x^n}, \, \, \, \, \, x \in [0,1].
$$ I want to show that $$\lim_{n \to \infty} \int_{0}^{1} f_n(x) dx = \int_{0}^{1} \left(\lim_{n \to \infty} f_n(x) \right) dx.$$ I know that the limit functio... | Note that $$\frac{x^n}{1+x^n}\leqslant 2$$ for all $x\in[0,1]$, and $$\int_0^1 2\ \mathsf dx = 2<\infty. $$
So by the dominated convergence theorem,
$$\lim_{n\to\infty}\int_0^1 f_n(x)\ \mathsf dx = \int_0^1 f(x)\ \mathsf dx = 0. $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1438318",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Evaluation of $ \int_{0}^{1}\left(\sqrt[4]{1-x^7}-\sqrt[7]{1-x^4}\right)dx$
Evaluation of $\displaystyle \int_{0}^{1}\left(\sqrt[4]{1-x^7}-\sqrt[7]{1-x^4}\right)dx$
$\bf{My\; Try::}$ We can write it as $$I = \displaystyle \int_{0}^{1}\sqrt[4]{1-x^7}dx-\int_{0}^{1}\sqrt[7]{1-x^4}dx$$
Now Using $$\displaystyle \bullet... | Hint: Notice that both integrands represent the same geometric shape, namely $X^4+Y^7=1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1438397",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Assume this equation has distinct roots. Prove $k = -1/2$ without using Vieta's formulas. Given $(1-2k)x^2 - (3k+4)x + 2 = 0$ for some $k \in \mathbb{R}\setminus\{1/2\}$, suppose $x_1$ and $x_2$ are distinct roots of the equation such that $x_1 x_2 = 1$.
Without using Vieta's formulas, how can we show $k = -1/2$ ?
Here... | The problem is find $k$ not $x_1$ isn't it? If so take the equation you arrived at before you started doing computer algebra:
$$(1-2k)[x_1 + (1/x_1)] - (3k+4) = 0$$
and multiply by $x_1$ to get
$$(1-2k)x_1^2 - (3k+4) x_1 + (1 -2k) = 0$$
but you know that
$$(1-2k)x_1^2 - (3k+4) x_1 + 2 = 0$$
so you must have $1 - 2k = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1438522",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find the value of : $\lim_{n\to \infty} \frac{1}{\sqrt{n}} \left(\frac{1}{\sqrt{n+1}}+\dotso +\frac{1}{\sqrt{n+n}} \right)$ I am trying to evaluate $\lim_{n\to \infty} \frac{1}{\sqrt{n}} \left(\frac{1}{\sqrt{n+1}}+\dotso +\frac{1}{\sqrt{n+n}} \right)$. I suspect identifying an appropriate Riemman sum is the trick. Howe... | This is a Riemann sum. Rewrite as
$$\frac1{n} \sum_{k=1}^n \frac1{\sqrt{1+\frac{k}{n}}} $$
which, as $n \to \infty$, becomes
$$\int_0^1 dx \frac1{\sqrt{1+x}} = 2 (\sqrt{2}-1)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1438637",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
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Why is this rotation "incorrect"? I've been trying to use the following formula for the rotation of a point around the origin:
$$
\begin{bmatrix}
x' \\ y'
\end{bmatrix} =
\begin{bmatrix}
\cos{\theta} & -\sin{\theta} \\
\sin{\theta} & \cos{\theta}
\end{bmatrix}
\begin{bmatrix}
x \\ y
\end{bmatrix}
$$
Now, I'm trying to ... | It is correct. What makes one think that it is wrong? You mean the convention that CCW is positive?
EDIT1:
What I meant is with Clockwise rotation I obtain $(-3,5)$ by matrix multiplication and Counterclockwise rotation gives $(3,-5)$ for a diametrically opposite point. So only association with sign convention of rota... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1438708",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Calculation of a covariant derivative and exterior derivative
I am reading these notes on differential geometry from a course at MIT. I have been verifying the computations for myself and I have a concern about the expression for $de^k$ on the final line. When I evaluate $de^k(e_i, e_j)$ according to the lemma, I obta... | It is, I think, a bit misleading to call that a Christoffel symbols: in general if we have a local coordinate $(x^1, \cdots, x^n)$, then it gives you a coordinate frame
$$\frac{\partial}{\partial x^1}, \cdots, \frac{\partial}{\partial x^n}$$
and the Christoffel symbols is defined as (write $\partial_i$ for the coordin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1438803",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Calculating which two points are the closest together given a series of points. I'm in a position where I need to calculate the distance between two points to determine which is the closest point. Now, the proper formula to do this is:
$$distance = \sqrt{(x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2}$$
That's all well and good, bu... | You can remove the square root -- just compare squared distances, rather than actual distances.
The multiplications will take very little time compared to the square root, but you can avoid some of those, too.
Suppose the current minimum distance is $d$, and suppose we're checking the point $P_i = (x_i, y_i, z_i)$ to s... | {
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Every translation of $f$ has no radial part
Let $f\in L^1(\mathbb{R}^n)$. Define the radial part of $f$ as
$$f_0(x)=\int_{S^{n-1}} f(||x||\omega)\,d\omega$$ where $\,d\omega$ is
the normalised surface integral over $S^{n-1}$. Define translation of
$f$ as $\ell_xf(y)=f(y-x)$. Suppose that $(\ell_xf)_0\equiv 0$ fo... | Since the integral over every sphere is zero, it follows (by Fubini) that the integral over every ball is zero. Hence, $f=0$ at every Lebesgue point, which is a.e. point (see the Lebesgue differentiation theorem).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1439016",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find the area of a figure in a triangle Triangle $ABC$ in the figure has area $10$ . Points $D,E,$ and $F$, all distinct from $A,B,$ and $C$, are on sides $AB,BC,$ and $CA$ respectively, and $AD=2$ , $DB=3$ .
If triangle $ABE$ and quadrilateral $DBEF$ have equal areas,then what is that area?
Efforts made: I've tried t... | $[DBEF]=[ABE]$ is equivalent, by subtracting $[DBE]$ to both sides, to $[DEF]=[DEA]$.
These triangles share the $DE$-side, hence $[DEF]=[DEA]$ implies $DE\parallel AF$, so:
$$ \frac{BE}{BC}=\frac{BD}{BA}=\frac{3}{5} $$
and the area of $[BDE]$, consequently, equals $\frac{9}{25}[ABC]=\frac{18}{5}$. Since $[ABE]=\frac{5}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1439222",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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} |
minimal projections in matrix-algebras Consider $A=\{ \begin{pmatrix} T & 0 \\ 0 & T \end{pmatrix}: T\in M_2(\mathbb{C})\}\subseteq M_4(\mathbb{C})$ and $p= \begin{pmatrix} 1 & 0&0&0 \\ 0 & 0&0&0\\0 & 0&1&0\\ 0 & 0&0&0 \end{pmatrix}\in A.$
We defined $p$ to be a projection, if $p^2=p=p^*$. I have already shown that $p$... | Here the answer again: $p$ isn't minimal in $M_4(\mathbb{C})$, because consider $q= \begin{pmatrix} 1 & 0&0&0 \\ 0 & 0&0&0\\0 & 0&0&0\\ 0 & 0&0&0 \end{pmatrix}$. It is $q=q^*=q^2$ and $0\le q\le p$, but it isn't $p=q$, because p has a smaller image than p.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is there more to explain why a hypothesis doesn't hold, rather than that it arrives at a contradiction? Yesterday, I had the pleasure of teaching some maths to a high-school student. She wondered why the following doesn't work:
$\sqrt{a+b}=\sqrt{a}+\sqrt{b}$.
I explained it as follows (slightly less formal)
*
*For y... | Here's another explanation approach:
Suppose that $$\sqrt{a+b}=\sqrt a + \sqrt b$$
If this is true, then we can square both sides of the equation:
$$(\sqrt {a+b})^2=a+b=(\sqrt a+\sqrt b)^2=a+b+2\sqrt {ab}$$
This resulting equality can then be manipulated as follows:
$$a+b=a+b+2\sqrt{ab}\\
0=2\sqrt {ab}\\
0=ab$$
By th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1439406",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "24",
"answer_count": 15,
"answer_id": 13
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Solving the nonlinear Diophantine equation $x^2-3x=2y^2$ How can I solve (find all the solutions) the nonlinear Diophantine equation $x^2-3x=2y^2$?
I included here what I had done so far. Thanks for your help.
Note: The equation above can be rewritten into $x^2-3x-2y^2=0$ which is quadratic in $x$. By quadratic formul... | I'll solve it in integers instead. $x^2-3x=2y^2$, by the quadratic formula, is equivalent to
$$x=\frac{3\pm\sqrt{9+8y^2}}{2}$$
So i.e. the problem is equivalent to solving $8y^2+9=m^2$ in integers.
$8y^2\equiv m^2\pmod{3}$, so $(y,m)=\left(3y_1,3m_1\right)$ for some $y_1,m_1\in\Bbb Z$, because $h^2\equiv \{0,1\}\pmod{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1439519",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 0
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Probability of drawing an ace Lets say I've drawn 20 cards from a deck and The 20th was an ace. I'm trying to figure out the probability of drawing an ace in the 21st draw.
I'm sure this is just clever combinatorics but all my attemps have led nowhere. The probability of The first event is 4/52, right? And the second o... | This problem can be solved with some conditional probability, though the method I will present is tedious and is probably not the most elegant.
We need to find $P(21st\ card\ is\ an\ Ace | 20th\ card\ is\ an\ Ace)$
We know this is equivalent to $\frac{\sum\limits_{i=2}^4 P(21st\ card\ is\ ith\ Ace\ \cap\ 20th\ card\ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1439613",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why $H^2U=UH^2$ implies $H$ and $U$ commutes? Why does $H^2U=UH^2$ imply $H$ and $U$ commutes, where $H$ is a Hermitian matrix and $U$ is a unitary matrix?
This comes from the book 'Theory of Matrices' on p277 http://www.maths.ed.ac.uk/~aar/papers/gantmacher1.pdf
Now I know the implication is false. How to prove the fo... | That cannot be true. Take
$$
H=\left(
\begin{array}{cc}
0 & 1 \\
1 & 0 \\
\end{array}
\right)
$$
then $H^2=1$ it commutes with anything, however
$$U=\left(
\begin{array}{cc}
0 & -i \\
i & 0 \\
\end{array}
\right)$$
is unitary and does not commute with $H$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1439766",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Application of Differentiation Prove that if the curve $y = x^3 + px + q$ is tangent to the x-axis, then
$$4p^3 + 27q^2 = 0$$
I differentiated $y$ and obtained the value $3x^2 + p$. If the curve is tangent to the x-axis, it implies that $x=0$ (or is it $y = 0$?). How do I continue to prove the above statement? Thanks.... | Notice, we have $$y=x^3+px+q$$ $$\frac{dy}{dx}=3x^2+p$$ Since, the x-axis is tangent to the curve at some point where $y=0$ & slope $\frac{dy}{dx}=0$ hence, we have $$\left(\frac{dy}{dx}\right)_{y=0}=0$$ $$3x^2+p=0\iff x^2=\frac{-p}{3}\tag 1$$
Now, at the point of tangency with the x-axis we have $$y=0\iff x^3+px+q=0$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1439876",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Solve for $x$ and represent the solutions on the trig circle? $$\sin(4x)\cos(x)+2\sin(x)\cos^2(2x) = 1- 2\sin^2(x)$$
I'm confused.
| Since $1-2\sin^2(x)=\cos (2x)$ and $\sin (2\alpha)=2\sin (\alpha)\cos (\alpha)$ we have
$$2\sin(2x)\cos(2x)\cos(x)+2\sin(x)\cos^2(2x)=\cos(2x)$$
$$\cos (2x)[2\sin(2x)\cos(x)+2\sin(x)\cos(2x)-1]=0$$
Then either $\cos (2x)=0$ or $2\sin(2x)\cos(x)+2\sin(x)\cos(2x)-1=0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1439980",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How does $-(-1)^n$ become $(-1)^{n+1}$ I am currently studying how to prove the Fibonacci Identity by Simple Induction, shown here, however I do not understand how $-(-1)^n$ becomes $(-1)^{n+1}$. Can anybody explain to me the logic behind this?
| The powers of $-1$ are $$-1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, \ldots$$ If $n$ is odd, then $(-1)^n = -1$, but if $n$ is even then $(-1)^n = 1$. So if you multiply $(-1)^n$ once by $-1$, it is the same as incrementing $n$. (Technically you can also decrement, but you can't generalize that to other negative n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1440106",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Prove if $P(A | B^c) = P(A | B)$ then the events A and B are independent. So I've started by saying that since $P(A | B^c) = P(A | B)$ we know that $\frac{P(A \cap B^c)}{P(B^c)} = \frac{P(A \cap B)}{P(B)}$. However I'm not sure where to go from there. Any help would be great!
| The defining relation is: $$P(A\cap B)=P(A|B)P(B)$$
You could also write: $$P(A\mid B)=\frac{P(A\cap B)}{P(B)}$$ but that asks for an apart treatment of the special case $P(B)=0$.
Let's say that $P(A\mid B)=c=P(A\mid B^c)$.
Then:
$$P(A)=P(A\mid B)P(B)+P(A\mid B^c)P(B^c)=cP(B)+cP(B^c)=c=P(A\mid B)$$
So: $$P(A\cap B)=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1440281",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Troubles with solving $\sqrt{2x+3}-\sqrt{x-10}=4$ I have been trying to solve the problem $\sqrt{2x+3}-\sqrt{x-10}=4$ and I have had tons problems of with it and have been unable to solve it. Here is what I have tried-$$\sqrt{2x+3}-\sqrt{x-10}=4$$ is the same as $$\sqrt{2x+3}=4+\sqrt{x-10}$$ from here I would square bo... | Note that when you square something like $a\sqrt{b}$ you get $a^2b$.
Thus, you should get:
$\begin{align}
x^2-6x+9 &= 64(x-10)\\
x^2-6x+9 &= 64x-640\\
x^2-70x+649 &= 0\\
(x-11)(x-59) &= 0\\
\therefore \boxed{x=11,59}.
\end{align}$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Help understanding the geometry behind the maximum principle for eigenvalues. Context:
I am learning about the maximum principle, which states that
If $A$ is a real symmetric matrix and $q(x)=\langle Ax, x \rangle$, the following statements hold:
*
*$\lambda_1=\mathrm{max}_{||x||=1} q(x)=q(x_1)$ is the largest eigen... | The eigenvectors for a real symmetric matrix are a complete orthonormal set, that is we can take them a an orthonormal basis for $R^n$.So we can represent any $v$ on the unit $n$-sphere as $(v_j)_{1\le j \le n}$ in the sense that there is a unique $(v_j)_{1 \le j \le n}$ such that $v=\sum_j (v_j x_j)$, and furthermore ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1440473",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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A theorem for cubic-A generalization of Carnot theorem I found a theorem for cubic, the theorem is a generalization of the Carnot theorem for conic. I'm an electrical engineer, not a mathematician. I don't know how to prove this result.
Let ABC be a triangle, Let three points $A_1, A_2, A_3$ lie on $BC$, three points $... | Let $S$ be a hypersurface of degree $d$ of equation $\phi(x) = 0$ in some $k^n$, $a$ a point not on $S$. Consider a line $a + t v$ through $a$ that intersects $S$ in $n$ points given by the equation in $t$
$$\phi(a + t v ) = 0$$
Now expanding the LHS wr to the powers of $t$ we get
$$\phi(a) + \cdots + \phi_d(v) t^d =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1440544",
"timestamp": "2023-03-29T00:00:00",
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Evaluation of $\int_0^\infty \frac{1}{\left[x^4+(1+\sqrt{2})x^2+1\right]\cdot \left[x^{100}-x^{98}+\cdots+1\right]}dx$
Evaluation of $\displaystyle \int_0^\infty \frac{1}{\left[x^4+(1+\sqrt{2})x^2+1\right]\cdot \left[x^{100}-x^{98}+\cdots+1\right]}dx$
$\bf{My\; Try::}$ Let $$I= \int_{0}^{\infty}\frac{1}{\left[x^4+(1+... | I think your way of evaluating the integral is very short and elegant. Nevertheless, you can do partial fraction decomposition. You'll end up with (as long as I did not do any mistakes)
$$
\frac{1}{\sqrt{2+a}}\biggl[\arctan\Bigl(\frac{\sqrt{2-a}+2x}{\sqrt{2+a}}\Bigr)+\arctan\Bigl(\frac{\sqrt{2-a}-2x}{\sqrt{2+a}}\Bigr)\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1440641",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Transforming a set of equations from one set of variables to another? Assume $\theta_1,\dots,\theta_n$ are $n$ positive numbers such that $$\theta_1+\dots+\theta_n=1$$ Define $y_{ij}=\frac{\theta_i}{\theta_j}$ for all $i,j \in \{1,\dots,n\}$. Is there a way to transform the above equation in terms of $y_{ij}$.
| No. If you multiply all $\theta_i$ by the same positive number $c\neq 1$, then all $y_{ij}$ stay the same, but $\theta_1+\dots+\theta_n$ changes.
In more details: assume that you have a condition or a set of conditions on $y_{ij}=\frac{\theta_i}{\theta_j}$ which is satisfied if and only if $\theta_1+\dots+\theta_n=1$. ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Rugby Union - Sum Of Binomial Random Variables I've been cracking my head at this for a few days now. Suppose a Rugby Union match is expected to have 5.9 tries on average, and tries follow a Poisson distribution. A conversion follows a try, where each conversion follows a binomial distribution with probability "p". Can... | Yes, if tries occur with a Poisson arrival rate $\lambda$, and conversions follow each try if there is a Bernoulli success of rate $p$, then conversions occur with a Poisson arrival rate of $\lambda p$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1440898",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Evaluate limit without L'hopital $$\lim_{x\to2}\dfrac{\sqrt{x^2+1}-\sqrt{2x+1}}{\sqrt{x^3-x^2}-\sqrt{x+2}}$$
Please help me evaluate this limit. I have tried rationalising it but it just can't work, I keep ending up with $0$ at the denominator... Thanks!
| $$\displaystyle \lim_{x\rightarrow 2}\frac{\sqrt{x^2+1}-\sqrt{2x+1}}{\sqrt{x^3-x^2}-\sqrt{x+2}}\times \frac{\sqrt{x^2+1}+\sqrt{2x+1}}{\sqrt{x^2+1}+\sqrt{2x+1}}\times \frac{\sqrt{x^3-x^2}+\sqrt{x+2}}{\sqrt{x^3-x^2}+\sqrt{x+2}}$$
So we get $$\displaystyle \lim_{x\rightarrow 2}\left[\frac{x^2-2x}{x^3-x^2-x-2}\times \frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1441036",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Prove that $\int_0^\infty\frac{x\cos(x)-\sin(x)}{x\left({e^x}-1\right)}\,dx = \frac{\pi}2+\arg\left(\Gamma(i)\right)-\Re\left(\psi_0(i)\right)$ While I was working on this question, I've found that
$$
I=\int_0^\infty\frac{x\cos(x)-\sin(x)}{x\left({e^x}-1\right)}\,dx = \frac{\pi}2+\arg\left(\Gamma(i)\right)-\Re\left(\ps... | Too long for a comment: After simplifying the integrand with x, write the $($new$)$
numerator as $\bigg(1-\dfrac{\sin x}x\bigg)-(1-\cos x),~$ then, for the latter, use Euler's formula
in conjunction with an exponential substitution to $($formally$)$ express the second
integral as a sum of harmonic numbers of complex... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1441123",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
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Kuhn Tucker conditions, and the sign of the Lagrangian multiplier I was under the impression that under the Kuhn-Tucker conditions for a constrained optimisation, with inequality constraints the multipliers must follow a non-negativity condition.
i.e. $$\lambda \geq 0$$
Operating with complementary slackness with the c... | The reason that you get the multiplier condition $\lambda \leq 0$ rather than $\lambda \geq 0$ is that you have a minimization problem, as you correctly indicate.
Intuitively, if the constraint $g(x,y)\leq c$ is active at the point $(x^*,y^*)$, then $g(x,y)$ is increasing as you move from $(x^*,y^*)$ and out of the do... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1441214",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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Noetherian Rings and Ideals Associated with $\text{Ann}(M)$ If $M$ is a finitely generated $A$-module where $A$ is Noetherian and $I$ is an ideal of $A$ such that the support of $M$, $\mathrm{Supp}(M)$ is a subset of $V(I)$ the set of prime ideals containing $I$. How do i know if there exists an $n$ such that $I^n$ is ... | Hint. $V(J)\subset V(I)\implies \mathrm{rad}(I) \subset \mathrm{rad}(J)$, so $I \subset \mathrm{rad}(J)$. Now use that $I$ is finitely generated.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1441307",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Statistics dice question $6$ people, $A, B, C, D, E, F$ sit in a circle to play a dice game.
Rule of game: as you roll a die and get a number which is a multiple of $3$, give the die to the person on the right(counterclockwise). If the number is not a multiple of $3$, give it to the person on the left.
What is the pro... | Let A=0 B=1 C=2 D=3 E=4 F=5
You are at A (ie 1), you have 5 chances and each chance can be +1 with Probability X(=1/3) or -1 with Probability Y(=2/3) such that
v + w + x + y + z = 1
Since RHS is positive, therefore number of positive term on LHS > no of negative term, hence solution is of the form
(v + w + x) + (y + z)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1441363",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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A limit problem where $x\to 0$ I can't figure out how to compute this limit: $$\lim_{x\to 0}\dfrac{e^{3x^2}-e^{-3x^2}}{e^{x^2}-e^{-x^2}}$$ Any advice?
| $$\lim_{x\to 0}\dfrac{e^{3x^2}-e^{-3x^2}}{e^{x^2}-e^{-x^2}}$$
$$=\lim_{x\to 0}\dfrac{e^{3x^2}-1-\left(e^{-3x^2}-1\right)}{e^{x^2}-1-\left(e^{-x^2}-1\right)}$$
$$=\dfrac{3\cdot\lim_{x\to 0}\dfrac{e^{3x^2}-1}{3x^2}-\left(\lim_{x\to 0}\dfrac{e^{-3x^2}-1}{-3x^2}\right)(-3)}{\lim_{x\to 0}\dfrac{e^{x^2}-1}{x^2}-\left(\lim_{x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1441560",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 5
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The identity $x/y-y/x = (y+x)(y^{-1}-x^{-1}).$ Difference of perfect square asserts that the expression $y^2-x^2$ factorizes as $(y+x)(y-x)$. On the train home last night, I noticed a variant on this. Namely, that $x/y-y/x$ factorizes as $(y+x)(y^{-1}-x^{-1}).$ Explicitly:
Proposition. Let $R$ denote a ring, and consi... | Not as far as I know, but notice that your identity is
$$ (x+y)\frac{x-y}{xy} = (xy)^{-1}(x+y)(x-y) = \frac{x^2-y^2}{xy} = \frac{x}{y}-\frac{y}{x}, $$
so I'm not sure I'd say that it's really distinct from difference-of-two-squares.
As far as generalisation goes, you've got the old
$$ x^n-y^n = (x-y)(x^{n-1}+x^{n-2}y+\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1441639",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What mistakes, if any, were made in Numberphile's proof that $1+2+3+\cdots=-1/12$? This is not a duplicate question because I am looking for an explanation directed to a general audience as to the mistakes (if any) in Numberphile's proof (reproduced below). (Numberphile is a YouTube channel devoted to pop-math and this... | I've argued this with others unsuccessfully, but I'm a glutton for punishment.
What you are saying is
assume that
$$1 - 1 + 1 - 1 + 1 - 1 + \ldots$$ is a convergent infinite
series
and let $$S_1 = 1 - 1 + 1 - 1 + 1 - 1 + \ldots$$
The assumption that
$$ 1 - 1 + 1 - 1 + 1 - 1 + \ldots$$ is a real number is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1441723",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
"answer_count": 5,
"answer_id": 1
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Is this a valid proof of the squeeze theorem? I'm self-studying Spivak's calculus, and I have no way of checking my solutions. One of the problems asks for a proof of the squeeze theorem. Here is what I have figured:
Proof. Suppose there exist two functions $f(x)$ and $g(x)$ such that $(\forall x)(g(x) \geq f(x))$ and... | Why does no such $\delta$ exist? This is not immediately obvious. you need to explain it.
A second problem with your proof is that the squeeze theorem only assumes that $f_1$ and $f_3$ converge. That the squeezed function $f_2$ also has a limit is part of what must be proved. But your proof simply assumes this is so.
F... | {
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"url": "https://math.stackexchange.com/questions/1441780",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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If $f$ is integrable on $[0,1]$, and $\lim_{x\to 0^+}f(x)$ exists, compute $\lim_{x\to 0^{+}}x\int_x^1 \frac{f(t)}{t^2}dt$. If $f$ is integrable on $[0,1]$, and $\lim_{x\to 0}f(x)$ exists, compute $\lim_{x\to 0^{+}}x\int_x^1 \frac{f(t)}{t^2}dt$.
I'm lost about what the value is for this limit in the first place. How ca... | To make a guess, first consider the simplest kind of function for which the limit exists, say $f=1$. Then $\lim_{x\to 0^{+}}x\int_x^1 \frac{f(t)}{t^2}dt=\lim_{x\to 0^{+}}x(1/x-1)=1$.
So clearly, if $\lim_{x\to 0} f(x)=l$, then $\lim_{x\to 0^{+}}x\int_x^1 \frac{f(t)}{t}dt=l$.
Now let's prove this. Let's use the conditi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1441888",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
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How to solve simultaneous modulus / diophantine equations Does anyone know how i'd solve a set of equations like this?
$(x_0\%k_0)\%2=0$
$(x_0\%k_1)\%2=1$
$(x_0\%k_2)\%2=0$
$(x_0\%k_3)\%2=1$
$(x_1\%k_0)\%2=0$
$(x_1\%k_1)\%2=0$
$(x_1\%k_2)\%2=1$
$(x_1\%k_3)\%2=1$
I'm trying to find valid values of $x_i$ and $k_i$. I've... | I came up with a solution that I'm happy with so far, since it lets me find decent solutions that lets me move on with my reason for wanting to solve these equations.
I'm solving these equations:
$x_0\%k_0=0$
$x_0\%k_1=1$
$x_0\%k_2=0$
$x_0\%k_3=1$
$x_1\%k_0=0$
$x_1\%k_1=0$
$x_1\%k_2=1$
$x_1\%k_3=1$
I solve them by usi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1442238",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
if $x_{1},x_{2} $ is $nx-x^n=a$ two roots, show that $|x_{1}-x_{2}|<\dfrac{a}{1-n}+2$
Assmue that $n$ be postive integers,and $a$is real number,and the equation $$nx-x^n=a$$ has postive real roots $x_{1},x_{2}$,show that
$$|x_{1}-x_{2}|<\dfrac{a}{1-n}+2$$
By condition, I showed that
$$(x_{1}-x_{2})[n-(x^{n-1}_{1}+x... | Let's study the function $f(x)=nx-x^n$ for $x>0$. $f'(x)=n(1-x^{n-1})$ so there is a global maximum in $1$. The function is monotonic increasing in $[0,1]$ and decreasing in $[1,\infty]$. The function attains twice the same value $a$ if and only if $a\in [0,n-1]$, that is $x\in [0,\sqrt[n-1]{n}]$. So for any such $x_1,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1442320",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Give a counterexample to $\bigcup\limits_{t \in T} (A_t \cap B_t)= \bigcup\limits_{t \in T} A_t \cap \bigcup\limits_{t \in T} B_t$ Let $\{A_t\}_{t \in T}$ and $\{B_t\}_{t \in T}$ be two non-empty indexed families of set.
Find a counterexample to
$$\bigcup\limits_{t \in T} (A_t \cap B_t)= \bigcup\limits_{t \in T} A_... | To make it more concrete, perhaps, let $T=\{\alpha,\beta\}$, where $\alpha\neq\beta$. Now, let $A_\alpha=\{\alpha\}$, $A_\beta=\{\beta\}$, $B_\alpha=\{\beta\}$, and $B_\beta=\{\alpha\}$. Then we have that
$$
\bigcup_{t\in T}(A_t\cap B_t)=(A_\alpha\cap B_\alpha)\cup(A_\beta\cap B_\beta)=\varnothing\cup\varnothing=\varn... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1442382",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Help with radical logs. $\log_ 2\sqrt{32}$ I am having trouble understanding logarithms. Specifically I can't understand this equation.
$$\log_2{\sqrt{32}}$$
I know the answer is $\left(\frac{5}{2}\right)$ but I don't know how it is the answer. If somebody could please explain the process step by step for solving this... | Using Logarithmic formula
$$\bullet \log_{e}(m)^n = n\cdot \log_{e}(m)\;,$$ where $m>0$ and $$ \bullet \log_{m}(m) =1\;,$$ Where $m>0$ and $m\neq 1$
So $$\displaystyle \log_{2}\sqrt{32} = \log_{2}(32)^{\frac{1}{2}} =\log_{2}\left(2^5\right)^{\frac{1}{2}}= \log_{2}(2)^{\frac{5}{2}} = \frac{5}{2}\log_{2}(2) = \frac{5}{2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1442473",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Isomorphism between subspaces Let $G,H$ be subspaces of a vector space $I$ such that $G \cap H = \{0\}$. Prove that $G + H \cong G\times H$.
I've defined $\phi: G + H \to G\times H$ as $\phi(g + h) = (g,h)$. Showing bijectivity as:
*
*$\phi(g + h) = \phi(j + k) \implies (g,h) = (j,k) \implies g = j, h = k$, so $g + ... | You have a short exact sequence:
\begin{alignat*}4
0\longrightarrow G\cap H && \longrightarrow G\oplus H &\longrightarrow G+H \longrightarrow 0 \\
x&&\longmapsto (x,x)\\
&&(x,y)&\longmapsto x-y
\end{alignat*}
As $G\cap H=\{0\}$ by hypothesis, the map $\; G\oplus H \longrightarrow G+H $ is an isomorphism.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1442577",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Proving $[(p\leftrightarrow q)\land(q\leftrightarrow r)]\to(p\leftrightarrow r)$ is a tautology without a truth table I came across the following problem in a book:
Show that if $p, q$, and $r$ are compound propositions such that $p$ and $q$ are logically equivalent and $q$ and $r$ are logically equivalent, then $p$ a... | Here is a proof which is slightly more algebraic than the "just look at the truth tables" one while not ballooning out into something lengthy and potentially unreadable:
\begin{array}l
(p \leftrightarrow q) \land (q \leftrightarrow r) \\
= \{\mbox{definition of $\leftrightarrow$}\} \\
(p \rightarrow q) \land (q \righta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1442870",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Why Dominated Convergence Theorem is not applicable in this case? Suppose $\omega$ is distributed uniformly over $(0,1]$. Define random variables $$X_n:=n\mathbf{1}_{(0,1/n]}.$$
Obviously, $X_n\rightarrow X=\mathbf{0}$ and $\lim_{n}E[X_n]$ is not equal to E[X]. In this case the Dominated Convergence Theorem (DCT) is no... | Consider $f=\max\limits_{n\in\mathbb{N}}n\mathbf{1}_{(0,1/n]}$, which is given by
$$
f(x)=\begin{array}{}n&\text{if }x\in\left(\frac1{n+1},\frac1n\right]\end{array}
$$
This would be the smallest candidate for a dominating function.
However, notice that
$$
\int_{\frac1{n+1}}^{\frac1n}f(x)\,\mathrm{d}x=\frac1{n+1}
$$
Thu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1442989",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Why is $f$ constant in this case? Reference: Conway - Functions of one complex variable p.55 Exercise 17.
Let $\epsilon>0$.
Let $f:B(0,\epsilon)\rightarrow \mathbb{C}$ be an analytic function such that $|f(z)|=|f(w)|$ for each $z,w\in B(0,\epsilon)$.
In this case, how do I prove that $f$ is constant?
I'm completely los... | Suppose that $f$ is not constant. Then $f'$ is not identically zero. Suppose, for instance (and without loss of generality), that $f'(0)\neq 0$ and $f(0)=1$. Now use the definition of the derivative to estimate $f(z)$ when $z$ is very close to $0$. Can you show there must be some $z$ such that $|f(z)|\neq 1$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1443059",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
Given a Fourier series $f(x)$: What's the difference between the value the expansion takes for given $x$ and the value it converges to for given $x$? The question given to me was: Find the Fourier series for $f(x) = e^x$ over the range $-1\lt x\lt 1$ and find what value the expansion will have when $x = 2$?
The Fourier... | That answer is wrong.
Define the function $f$ to be $\exp(x)$ on $[-1, 1)$ and continue it periodically. Then by the Dirichlet condition the Fourier series of $f$ converges for all $x$ to $\frac{1}{2}(f(x+) + f(x-))$. Since the function has a jump-type discontinuity at $x = 1$, the Fourier series at this point will co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1443124",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
Proving that if $\lim_{n \rightarrow \infty} a_n = a$ then $\lim_{n \rightarrow \infty} a_n^2 = a^2$ If we have a real sequence $\left|a_n\right|$ such that $\lim_{n \rightarrow \infty} a_n = a$, how do we prove (by an $\epsilon - N$ argument) that $\left|a_n\right|$ such that $\lim_{n \rightarrow \infty} a_{n}^{2} = ... | If for every $\varepsilon > 0$ there is some $N \geq 1$ such that $|a_{n}-a| < \varepsilon$, then, taking any $\varepsilon > 0$, there is some $N \geq 1$ such that
$$
|a_{n}^{2}-a^{2}| = |a_{n}-a||a_{n}+a| < 2|a|\varepsilon+\varepsilon^{2}
$$
for all $n \geq N$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1443190",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
How can you algebraically determine if a curve is symmetric about $y=-x$? If I have a curve implicitly defined by say $x^2+xy+y^2=1$, then it is clear that it is symmetric about $y=x$ because if I interchange x's with y's, then I have the exact same equation.
However, how would one adapt a similar kind of mentality to ... | Symmetry in line $y=-x$ maps point $(x,y)$ to $(-y,-x)$ (and vice versa), so you indeed can simply replace $x$ by $-y$, $y$ by $-x$ and check whether you get the same equality.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1443270",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Playing with closure and interior to get the maximum number of sets Can you find $A \subset \mathbb R^2$ such that $A, \overline{A}, \overset{\circ}{A}, \overset{\circ}{\overline{A}}, \overline{\overset{\circ}{A}}$ are all different?
Can we get even more sets be alternating again closure and interior?
| According to Kuratowski's closure-complement problem, the monoid generated by the complement operator $a$ and the closure operator $b$ has $14$ elements and is presented by the relations $a^2 = 1$, $b^2 = b$ and $(ba)^3b = bab$. Now you are interested by the submonoid generated by the closure operator $b$ and by the in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1443348",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
How do you explain that in $y=x^2$, y is proportional to the square of x? My understanding is that all proportional relationships are linear relationships. If this is indeed the case, how is it that we can also say that in a non linear equation like $y = x^2$, y is proportional to $x^2$?
| The definition of proportionality is that $A \propto B \iff A = kB$. So A is linearly related to B, and also $A=0$ iff $B=0$.
In your case, $y = (1) x^2$ so y is actually proportional to $x^2$.
I think you are confused by "y is linear to x squared". Actually linear means that the power is 1. But when I say that y is li... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1443445",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 0
} |
Can't calculate an eigenvector because the system has no solutions. Find all eigenvalues of the matrix $A=\begin{pmatrix}
2&2&-2\\
2&5&-4\\
-2&-4&5\\
\end{pmatrix}$
and find matrix $U$ such that $U^TU=I$ and $U^TAU$ is diagonal.
I calculated and checked that $\lambda_1=10,\lambda_{2,3}=1$. ... | What you need is a second (linearly independent) eigenvector associated with $\lambda = 1$. That is, you need to find another solution $x$ to
$$
(A - I)x = 0
$$
What you were looking for was a solution to
$$
(A - I)x = v_1
$$
which would be a generalized eigenvector. Generalized eigenvectors are only useful when th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1443539",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Boundedness of a set Let $$S = \{(x, y) \in \mathbb{R}^2: x^2 + 2hxy + y^2 =1\}$$
For what values of $h$ is the set $S$ nonempty and bounded?
For $h = 0,$ it is surely bounded, the curve being the unit circle. What for other $h$?
Please help someone.
| Hint:
$x^2+2hxy+y^2-1=0$ is a conic, and it is an ellipse if its discriminant $B^2-4AC=4(h^2-1)$ is negative.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1443649",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Looking for Taylor series expansion of $\ln(x)$ We know that the expansion of $$\sin(x)
$$ is $$x/1!-x^3/3!\cdots$$
Without using Wolfram alpha, please help me find the expansion of $\ln(x)$.
I have my way of doing it, but am checking myself with this program because I am unsure of my method.
| $f(x)=ln(x)$
taylor expansion around a is
$f(x)=\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$
so for $a = 1$
$f'(x) = \frac{1}{x}$ $f''(x) = \frac{-1}{x^2}$ $f^{(3)}(x) = \frac{2}{x^3}$
$f^{(n)}(x) = \frac{(n-1)! (-1)^{n-1} }{x^n}$
$f(x)=\sum_{n=0}^{\infty} \frac{f^{(n)}(1)}{n!}(x-1)^n=\sum_{n=1}^{\infty} \frac{(-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1443717",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Behaviour of $f(x)=e-\left(1+\frac{1}{x} \right)^{x}$ when $x\to+\infty$ This is from an MCQ contest.
For all $x\geq 1$ let $f(x)=e-\left(1+\dfrac{1}{x} \right)^{x}$ then we have :
*
*$f(x)\mathrel{\underset{_+\infty}{\sim}}\dfrac{e}{x}$ and $f$ is integrable on $[1,+\infty[$
*$f(x)\mathrel{\underset{_+\i... | Answers 1. and 3. can be excluded a priori, because if $f(x)\sim k/x$ then $f$ cannot be integrable on $[1,+\infty[$.
Then it is quite obvious that we can make a Taylor expansion in $1/x$, which will never yield a $\sqrt{x}$ term: if so 2. can be excluded and the correct answer should be 4.
In fact, using $\ln(1+t)=t-t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1443786",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove that as a $\Bbb Q$ vector space $\Bbb R^n$ is isomorphic to $\Bbb R$
Prove that as a $\mathbb{Q}$ vector space, $\mathbb{R}^n$ is isomorphic to $\mathbb{R}.$
I came across this statement somewhere and I have been trying to prove it ever since, but I just don't know how to start or what to define as the map.
| You can use the statements that both $\Bbb R$ and $\Bbb R^n$ are infinite (uncountable) dimensional extensions over $\Bbb Q$ and two vector spaces of same dimension are isomorphic. And $|\Bbb Q| < |\Bbb R|$. So $dim(\Bbb R)= |\Bbb R|= |\Bbb R^n|=dim(\Bbb R^n)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1443871",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
Is there a name for the rule $a \div (b \times c) = a \div b \div c$? Edit, because I should have looked it up before I posted the question:
Is there a name for the rule $a \div (b \div c) = a \div b \times c$ ? I ran across this in Liping Ma's book, Knowing and Teaching Mathematics, and I have searched the internet f... | I suppose that $a,b,c$ are rational (or real) numbers. In this case your starting expression is equivalent to:
$$
\dfrac{a}{b\times c}=a\times \dfrac{1}{b}\times \dfrac{1}{c}=\left(a \times \dfrac{1}{b} \right)\times \dfrac{1}{c}=\dfrac{a \times \dfrac{1}{b}}{c}=\dfrac{ \dfrac{a}{b}}{c}
$$
so you can see that this prop... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1443978",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 0
} |
Prove that $|x+y+z| \le |x|+|y| +| z|$ for all $x, y, z$
Prove that $$\lvert x+y+z \rvert \le \lvert x \rvert + \lvert y \rvert + \lvert z\rvert $$ for all $x, y, z$.
| You can prove it by using triangle inequality of two numbers.Consider $(y+z)$ as a single entity,$|x+(y+z)|\le |x|+|y+z| $, now use triangle inequality on $|y+z|$, and you will get your answer.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1444046",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Writing $\sqrt[\large3]2+\sqrt[\large3]4$ with nested roots Let $C\subset\Bbb R$ be the smallest set containing $0$ and closed under whole number addition/subtraction, whole number exponents, and whole number roots. That is, for all $c\in C$ and $n\in\Bbb N$, we have $c\pm n\in C$, $c^n\in C$, and $c^{1/n}\in C$.
We kn... | Note
$$(\sqrt[3]2+\sqrt[3]4 -1)^2= 5- \sqrt[3]{4}
$$
which yields
$$\sqrt[3]2+\sqrt[3]4 =\sqrt{5-\sqrt[3]{4}}+1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1444163",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
How to prove $f(n)=\lceil\frac{n}{2}\rceil$ is one-to-one and onto? Edit: Both domain and codomain are the set of all integers ($\mathbb{Z}$)
I proved $f(n) = \lceil\frac{n}{2}\rceil$ was one-to-one this way:
$\forall{x_1}\forall{x_2}[(f(x_1) = f(x_2)) \to(x_1 = x_2)]$
Let $f(n) = f(x) = y$
Let $x_1 = 2$ and Let $x_2 ... | In general when you prove functions are 1-1 and onto, you need to start by specifying the domain and codomain for your function. In this case both your domain and codomain are $\mathbb{Z}$, so you can write your ceiling function as $f:\mathbb{Z} \to \mathbb{Z}$.
To prove $f$ is onto, you want to show for any number $y$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1444281",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
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