Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
How can I solve this differential equation (not a obvious 2nd order ODE) I am not sure how to manipulate this 2nd order ODE to find the general solution
$$u_r u^{'}+\dfrac{u_r}{r}u=\dfrac{\nu}{r}\left(\dfrac{d}{dr}\left(r u^{'}\right) - \dfrac{u}{r}\right)$$
Where $$u_r=-\dfrac{r_0 v_w}{r}$$ where $r_0$ and $v_w$, $\nu... | The usage of abbreviation $u_r$ is somewhat misleading. Instead, I would suggest to do as follows. Let us denote constant $r_0 v_w$ as $\mu$. Then your equation can be rewritten as follows:
$$
-\frac{\mu}{r^2}\left(ru'+u\right) = \frac{\nu}{r}\left(ru''+u'-\frac{u}{r}\right).
$$
After simple manipulations, we get
$$
u'... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1985613",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
how to find a Borel function $f$ such that $a(t) = f(b(t))$ Let $b(t)$ be a non-increasing, right-continuous and positive function on $(0,\infty)$ (clearly, it is Borel) and $a(t)$ be a measurable function on $(0,\infty)$. Moreover, $a(t)$ is a constant on an interval when $b(t)$ is a constant on this interval.
Can we ... | In general, no: if $L$ is the Cantor function, let $b=2-L$ and $a=2+1_{S}$, where $1_S$ is the indicator function of a non-Borel subset $S$ of the Cantor set $C$. Then $a$ is non-Borel and constant on all the intervals where $b$ is constant (i.e. on $\Bbb R\setminus C$), but the RHS you require is clearly a Borel funct... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1985740",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Show that $|z| = 1$ if and only if $\bar{z} = \frac{1}{z}$. Maybe a very stupid question but I am stuck. Show that $|z| = 1$ if and only if $\bar{z} = \frac{1}{z}$.
Is it enough to simply multiply, i.e. $z\bar{z} = \frac{1\times z}{z} = 1$? Showhow I feel this is not correct. I know that if $z = \pm 1$ or $z \pm i$ the... | Take $z$ in the unit circle. $\bar{z}$ is the reflection of $z$ with respect to the real axis. Therefore, $\bar{z}$ has modulus $1$ and argument the negative of the argument of $z$. Since we multiply complex numbers by multiplying their modulus and adding their arguments, we have the $z\bar{z}$ has modulus $1$ and argu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1985829",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 1
} |
Calculate the sum of the series $\sum_{n=1}^\infty \frac 1{n(n+1)^2}$ I have already proven that
$$\int_0^1 \ln(1-x)\ln (x)\mathrm d x=\sum_{n=1}^\infty \frac 1{n(n+1)^2}.$$
So now I want so calculate
$$S:=\sum_{n=1}^\infty \frac 1{n(n+1)^2}.$$
I think I should use the fact that
$$\sum_{n=1}^\infty \frac 1{n^2}=\fra... | we have that
$$\sum_{n=1}^{m }\frac{1}{n}- \frac{1}{n+1}=\frac{m}{m+1}=1-\frac{1}{m+1}$$ the sum is $1$when the $m\rightarrow \infty$
$$\sum_{n=1}^{\infty }\frac{1}{n(n+1)^2} = \sum_{n=1}^{\infty }(\frac{1}{n}- \frac{1}{n+1}) -\sum_{n=1}^{\infty } \frac{1}{(n+1)^2}=1 -\sum_{n=2}^{\infty } \frac{1}{n^2}$$
$$=1-(\frac{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1986102",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
$x + y = 7 $and $x^3 - y^3 = 37$, find $xy$
$x+y=7$
$x^3-y^3 = 37$
Find $xy$
I have tackled this one but I am stuck, when I get to this point:
$$(x-y)((x+2)^2 -xy) = 37$$
I would be glad if you could give me any suggestions.
| Since $(x^3-y^3)^2$ is symmetric in $x$ and $y$, you can express it as a polynomial in terms of $x+y$ and $xy$ :
$(x^3-y^3)^2 = x^6 - 2x^3y^3 + y^6 = (x+y)^6 - (6x^5y+15x^4y^2+22x^3y^3+15x^2y^4+6xy^5) \\
= (x+y)^6 - 6xy(x+y)^4 + (9x^4y^2+14x^3y^3+9x^2y^4) \\
= (x+y)^6 - 6xy(x+y)^4 + 9(xy)^2(x+y)^2 - 4(xy)^3$
Plugging t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1986213",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
Prove that if $f : X \to Y$ is a function between non-empty finite sets such that $|X| \lt |Y|$, then $f$ is not a surjection.
Theorem 11.1.6: Suppose that $f: X \to Y$ is a function between non-empty finite sets such that $|X| \lt |Y|$. Then $f$ is not a surjection, i.e. there exists an element of $Y$ which is not a ... | This proof is correct.
Normally people would phrase this argument in a more informal manner,
along the lines of: "If $f: X \to Y$ is surjective, and $X$ has $n$ elements,
then $Y$ has $\leq n$ elements."
But in the context of your particular text and course, your argument looks good.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1986426",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Classifying permutations in terms of their cycle notation Is there are a standard way of referring to permutations in terms of their cycle notation? For example: Does the set of all permutations in $S_4$ that can be expressed as the composition of two two-cycle permutations $\left\{ (12)(34), (13)(43), (14)(23)\right\}... | The sets you're looking for are the conjugacy classes of $S_n$.
It's a good exercise to convince oneself that "$a$ and $b$ are conjugates in $S_n$" is equivalent to "$a$ and $b$ have the same cycle type".
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1986546",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Prove the following series converges Let $u, v, w$ be real numbers such that $u+v+w=0$.
Suppose that $\{b_k\colon k=0,1,2,\dots\}$ is a sequence of real numbers such that $\lim_{k\to\infty} b_k=0$.
For $k=0,1,2,\dots$ define
$a_{3k}=u b_k,$ $a_{3k+1}= v b_k,$ $a_{3k+2} = w b_k.$
Prove that the series $ \sum_{n=0}^{\in... | Let us have $c_{3k}=u,\ c_{3k+1}=v,\ c_{3k+2}=w$. By the Dirichlet test, we need to show that $\sum_{k=0}^N c_k$ is bounded and that $\lim_{n\to\infty}b_n=0$.
Clearly, we have
$$\sum_{k=0}^N c_k\le\sum_{k=0}^Mc_k$$
for some $M\in[0,1,2]$ and we already have $\lim_{n\to\infty}b_n=0$, meaning that
$$\sum_{k=0}^\infty c_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1986648",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Statistical significance in the interval The statistical significance of an observed result is not always the same as the material significance - the practical importance - of that result.
I am curious as to why this would be the case in some circumstances. And is there an example when a statistically significant resul... | Statistically significant, not of practical importance. I once consulted on a study to determine whether a new machine and
a standard lab test gave the same results for hemoglobin (Hgb) in the blood.
Hgb was determined for about three dozen newborn babies using both methods
for each. The purpose of the study was to see... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1986772",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that $\cos^2(z)+\sin^2(z)=1$ Can anybody show me how to prove that $\cos^2(z)+\sin^2(z)=1$ for $z\in \mathbb{C}$? I can prove it for the case where $z$ is real, but can't seem to find a way to prove it for complex numbers in general.
| The very same proof that works for real values works for complex values: both $\sin z$ and $\cos z$ admit convergent powerseries on the whole complex plane, which you can differentiate term by term, giving that
$$(\cos z) ' = -\sin z$$
$$(\sin z)' = \cos z$$
If you consider $f(z) = \cos^2 z +\sin^2 z-1$ the above gives... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1986878",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Uniqueness for a linear transformation given nullspace and range Find the representation in a canonical base for: $$\mathcal{T}: \mathbf{R^3} \rightarrow \mathbf{R^3}$$
Such that :$$ \quad\mathcal{N}(\mathcal{T}) = \quad \langle\left(\begin{matrix} 1\\0\\-1 \end{matrix}\right)\rangle$$ and the image of $\mathcal{T}$ is... | Suppose we have a linear map from $T:U\rightarrow V$ and we have a basis for $U$. If I tell you where $T$ takes each basis vector of $U$ then you know $T$ exactly, and therefore these conditions define $T$ uniquely. Now suppose instead of telling you where specific basis vectors go, I just tell you a vector in $U$ gets... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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How to prove or disprove this lemma? Lemma: For any real-valued continuous function $ f(x,y) $, where $ x \in R^m $ and $ y \in R^n $.
There are smooth scalar functions $ A(x)\geq 0 $, $ B(y)\geq 0 $, $ C(x)\geq 1 $ and $ D(x)\geq 1 $ such that
$ \left| f(x,y) \right| \leq A(x)+B(y) $
$ \left| f(x,y) \right| \leq C(x)D... | Let $M(N) = \max \{ |f(x,y)|: \max(\|x\|,\|y\|) \le N+1 \}$ (which is finite because $\{(x,y): \max(\|x\|,\|y\|) \le N\}$ is compact). There are smooth increasing functions $a(x)$ and $a(y)$ such that $a(n) > M(n)$ and
$b(n) > M(n)$ for each nonnegative integer $n$. Take $A(x) = a(\|x\|)$ and $B(y) = b(\|y\|)$ (that... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1987119",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that $a^2$ when divided by $5$ cannot have a remainder of $3$. For any natural number $a$, prove that $a^2$ when divided by $5$ cannot have a remainder of $3$. (Hint: What are the possible values of the remainder when $a$ is divided by $5$?
Using the hint, I found that the possible values of the remainder are $1,... | if $a \equiv 0 \mod 5$, then $a^2 \equiv 0 \mod 5$.
if $a \equiv 1 \mod 5$, then $a^2 \equiv 1 \mod 5$.
if $a \equiv 2 \mod 5$, then $a^2 \equiv 4 \mod 5$.
if $a \equiv -2 \mod 5$, then $a^2 \equiv 4 \mod 5$.
if $a \equiv -1 \mod 5$, then $a^2 \equiv 1 \mod 5$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1987223",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
When are $\beta_1, \beta_2, \ldots, \beta_n$ linear independent? Given $n$ linear independent vectors, $\alpha_1, \alpha_2, \ldots, \alpha_n$.
Now, let
$$\beta_1 = \alpha_1 + \alpha_2 + \cdots + \alpha_m$$
$$\beta_2 = \alpha_2 + \alpha_3 + \cdots + \alpha_{m+1}$$
$$\ldots$$
$$\beta_{n-m} = \alpha_{n-m} + \alpha_{n-m+1}... | Isn't the implied matrix a circulant matrix?
Then, because of this determinant formula, your problem (for the case of linearly dependent vectors) appears to ask, for given $m < n$
When is it the case that there is a $j \mid n$ such that $\omega_{j}$ is a root of $f_{m} = x^{m-1} + \dots + x + 1$, or equivalently $\phi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1987341",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
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Prove: $ \frac{1}{\sin 2x} + \frac{1}{\sin 4x } + \cdots + \frac{1 }{\sin 2^n x} = \cot x - \cot 2^n x $
Prove
$$ \frac{1}{\sin 2x} + \frac{1}{\sin 4x } + \cdots + \frac{1 }{\sin 2^n x} = \cot x - \cot 2^n x $$
where $n \in \mathbb{N}$ and $x$ not a multiple of $\frac{ \pi }{2^k} $ for any $k \in \mathbb{N}$.
My... | $$\csc2A+\cot2A=\dfrac{1+\cos2A}{\sin2A}=\cot A$$
$$\iff\csc2A=\cot A-\cot 2A$$
Put $2A=2x,4x,8x,\cdots,2^nx$ and add
See also: Telescoping series
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1987415",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
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Periodic functions and properties of integration This is from a book in Fourier series.
I don't understand how the integral can be split as in equality 2 and 3.
Which properties of integration justify that?
$F(x)$ and $f(x)$ are two functions with the periodicity $2l$. The integral $F(x)=\int_{-l}^x f(t) \, \mathrm d... | Let us consider the function f (x)=cos x.
At x=-phi and+phi f (x) is zero.
Here,l=phi and let us consider x=phi/6.
When considered the area of f (x) from -phi to 13phi/6.
In the region 0 to phi the curve area is positive.
In the region phi to 2phi the curve area is negative.
Now cosider the area from -phi to 13phi/6.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1987557",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proj as a quotient of an affine variety by $\mathbb C^*$ I wonder how good is the quotient of $\operatorname{Spec} A$ by $\mathbb C^*$ given by $\operatorname{Proj} A$.
First, an action of the group $\mathbb C^*$ on an complex affine variety $\operatorname{Spec} A$ is the same as $\mathbb Z$-grading on its function rin... | Let's write $X:=\operatorname{Spec}(A)\subseteq\Bbb A^n\newcommand{\sms}{\mathrm{ss}}$ where I also assume that the coordinates of $\Bbb A^n$ are such that $\Bbb C^\times$ acts by a character on it (we can always choose coordinates in this way). Since you insisted that the induced grading vanishes in negative degrees, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1987659",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Very stuck with flow on a circle dynamical system problem I want to find the bifurcation points and be able to draw and classify the phase portrait as the parameter u varies for the following
$$\theta^{\bullet}=u+\sin(\theta)+\cos(2\theta)$$
(where $\theta^{\bullet}=\frac{d}{dt}\theta$)
but I am having trouble.
This i... | Consider the Jacobian of your system defined by $$\dot{\theta} = f(\theta) = u+\sin(\theta)+\cos(2\theta)$$ which we can rewrite in terms of $\sin(\theta)$ and $\cos(\theta)$ as $$f(\theta) = u+\sin(\theta)+\cos^2(\theta)-\sin^2(\theta)$$ The Jacobian is $$J(\theta) = \cos(\theta)-2\sin(2\theta) = \cos(\theta)-4\sin(\t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1987793",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Why does the Discrete Fourier Transform and the Fast Fourier Transform give different results? I have a function
$$
f(t) =
\begin{cases}
a(1-a|t|) &\text{if } |t|<\dfrac{1}{a} \\[6pt]
0 &\text{if } |t|>\dfrac{1}{a}
\end{cases}
$$
Here is what the function looks like for $a=2$:
I wrote two scripts to evaluate the Four... | The FFT returns you a signal in the range $[0-2f_\text{max}]$ where $f_\text{max}=1/(2dt)$. Also it gives you a signal where the negative frequencies come after the positive. To resolve that issue use 'fftshift' after using 'fft', you will see that the spectrum will become symmetric as you want it since your original ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1987907",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If $X_{1}, X_{2},..$ are identically distributed then $M_{n}/n \rightarrow 0$ in probability. If $X_{1}, X_{2},..$ are identically distributed then $M_{n}/n \rightarrow_{p} 0$ where $M_{n}=max\{|X_{1}|,...,|X_{n}|\}$ and $E(|X_{1}|)$ is finite.
$M_{n}/n \rightarrow_{p} 0$ by definition means given any $\epsilon$, $P(M... | $M_n>n\varepsilon$ if and only if $|X_i|>n\varepsilon$ for some $1\leq i\leq n$. Therefore
$$ \{M_n>n\varepsilon\}=\bigcup_{i=1}^n\{|X_i|>n\varepsilon\}$$
hence
$$ \mathbb{P}(M_n>n\varepsilon)\leq \sum_{i=1}^n\mathbb{P}(|X_i|>n\varepsilon)=n\mathbb{P}(|X_1|>n\varepsilon) $$
Therefore it is enough to show that
$$ \lim_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1988028",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is the alternating series $\sum_{n=1}^{\infty }(-1)^n \frac {n^2 - 1}{2n^2 + 3}$ divergent? I tried every test for convergence and really came up with nothing. Without answering the problem directly, is it possible to determine the divergence or convergence for this series?
$$\sum_{n=1}^{\infty }(-1)^n \frac {n^2 - 1}{... | Let
$$a_n=(-1)^n\frac{n^2-1}{2n^2+3}$$
we have $$\forall n\in \mathbb N^* \;\;
|a_n|=\frac{1-\frac{1}{n^2}}{2+\frac{3}{n^2}}$$
which yields to
$$\lim_{n\to +\infty}|a_n|=\frac{1}{2}$$
and
$$\lim_{n\to+\infty} a_n \neq 0$$
thus, the series $\sum a_n$ is divergent.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1988194",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Solving $\frac{3}{2} e^y y^2 + y = 0$ Trying to solve the following Eq, but none of the usual methods seem to work:
$$3/2 \exp{(y)}y^2+y=0$$
Any help is appreciated.
| If $ay^2e^y+y=0$,
then
$ye^y=-1/a$.
By the definition of the
Lambert W function
(see https://en.m.wikipedia.org/wiki/Lambert_W_function,
for example)
$y=W^{-1}(-1/a)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1988333",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Open superset of $\mathbb{Q}$ Let $S$ be an open set such that $\mathbb{Q}\subset S$. We can also define a set $T=\mathbb{R}\setminus S$. I have been trying to prove or disprove whether $T$ could be uncountable. I suspect $T$ has to be at most countable, is my intuition correct?
| Your intuition is reasonable, but incorrect. HINT: the rationals are countable, so we can list them as $\mathbb{Q}=\{q_1, q_2, q_3, . . .\}$ (obviously this listing isn't "in order," in any sense, but that's fine).
Now, let $U_n=(q_n-2^{-n}, q_n+2^{-n})$. Each $U_n$ is open, so the union $V=\bigcup U_n$ is open, and c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1988455",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 0
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Condition for pointwise convergence of $({f_n}')_{n\in\mathbb{N}}$ to $f'$ if $({f_n})_{n\in\mathbb{N}}$ converges pointwise to $f$ Problem
Let $(f_n)_{n\in\mathbb{N}}$ be a sequence of differentiable functions such that $f_n:A(\subseteq \mathbb{R})\to \mathbb{R}$ for all $n\in\mathbb{N}$. Let $(f_n)_{n\in\mathbb{N}}$... | Hint: The function $h(x) =\int_0^x \sin (1/t)\,dt$ is differentiable everywhere, with $h'(x) = \sin (1/x)$ for $x\ne 0$ by the FTC, and $h'(0)=0$ because of all the oscillation at $0.$ (You have to do some work to see this last bit. ) Now you have mentioned an example of a sequence $f_n$ converging to $0$ uniformly on ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1988673",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Showing an analytic function on the unit disk is nonzero in a certain neighborhood Suppose $f(z)$ is analytic for $|z|\le 1$ and $f(0) = a_0 \ne 0$. If $M = \max_{|z|=1} |f(z)|$, then show $f(z)\ne 0$ for all $z$ with $|z| < \frac{|a_0|}{|a_0|+M} =:r$.
I know we can write $f(z) = a_0 + z^kg(z)$, some $k\ge 1$ and $g$ a... | The map $g(z):=f(z)/M$ is a analytic map from the unit disc in itself.
Then by the Schwarz-Pick Theorem, for $|z|< 1$,
$$\left|\frac{g(z)-g(0)}{1-\overline{g(z)}g(0)}\right|\leq |z|.$$
If $f(z)=0$ and $|z|<r<1$ then $g(z)=0$ and we obtain
$$\frac{|a_0|}{M}=|g(0)|\leq |z| <r= \frac{|a_0|}{|a_0|+M}$$
which is a contradic... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1988771",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
4-digit password with unique digits not in ascending or descending order I need to calculate how many possible passwords there are if each password is 4 digits long, using the digits 0-9. All digits in the password must be unique, and cannot all be neither increasing nor decreasing. For example “3569” is not allowed, b... | As you have already worked out, there are $^{10}P_4=5040$ passwords that repeat no digit.
From this number we are to subtract those passwords whose digits are all increasing or all decreasing. All such passwords can be generated by picking four digits out of the ten without regards to order – there are $\binom{10}4=210... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1988872",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 2,
"answer_id": 0
} |
What's so discrete about discrete topology? I am a beginner at topology. I recently learned about discrete topology.
But the definition of discrete topology doesn't convey anything 'discrete' to me. Is it just whimsically named like top, bottom quarks in physics or does discreteness has anything to do with discrete top... | Do you understand what's so isolated about an "isolated" point? Well, a discrete space is a space in which every point is isolated. This seems to agree well with the dictionary definition of "discrete": "consisting of distinct or unconnected elements."
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1988978",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 3
} |
Non Linear PDE Using Charpit's Method . Using Charpit's method .
How Can I solve this :
$$u^{2}(p^{2}+q^{2})=x^{2}+y^{2}$$
where$$ du=p dx+q dy \ , p =u_{x} ,q=u_{y}$$
So we have
$$f(x,y,p,q,u)=u^{2}(p^{2}+q^{2})-x^{2}+y^{2}=0$$
and $$\frac{dx}{-2pU}=\frac{dy}{-2qU}=\frac{dU}{-2U(p^{2}+q^{2})}=\frac{dp}{-2x+p(p^2+q^2)}... | Use the change of variable
$$
X=x^2,\quad Y=y^2,\quad U=u^2\\
P=\frac{\partial U}{\partial X},\quad Q=\frac{\partial U}{\partial Y}
$$
Note that
$$
P=\frac{\partial U}{\partial u}\frac{\partial u}{\partial x}\frac{\partial x}{\partial X}=\frac{up}{x}\\
Q=\frac{\partial U}{\partial u}\frac{\partial u}{\partial y}\frac{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1989128",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Integration of $\cos(x)/(5+\sin(x)^2)$ Is the following integration correct?
Consider the integral $\int_{-\pi}^\pi \frac{\cos(x)}{5+\sin(x)^2} dx$.
Substitute $y = \sin(x)$ then we have $\frac{dy}{dx} = \cos(x)$ and hence $dy = \cos(x) dx$ and we get
$$\int_{-\pi}^\pi \frac{\cos(x)}{5+\sin(x)^2}dx = \int_{\sin(-\pi)}^... | Hint: split the integral into two pieces $-\pi$ to $0$ and $0$ to $\pi$. Note the point symmetry of the integrand for the points $P_1=(-\pi/2,0)$ and $P_2=(\pi/2,0)$. You can achieve this by substitution $x=u\pm \pi/2$ and $dx = du$. You should see that you will get odd functions after the substitutions, which both ev... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1989269",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
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Linear System of Equations
I've been given:
$$
\begin{array}{rcrcrcr}
x & + &2y &+& z &=& 3\\
x & +& y &+& 2z &=& 2\\
& & y&-& z &=& 1
\end{array}
$$
Let $S$ be the set of all triples that are solutions to the system above and $$T = {(x,y,z)}={(1-3t, 1+t, t)}
$$
I need to show that $S=T$.
So, I tried solvin... | *
*First of all, do you know how to solve a linear equation by Gaussian elimination?
*To show $T=S$, you could show $T\subset S$ and $S\subset T$. One of the direction is trivial, can you see why?
*Note that
$$
T = \{(1-3t, 1+t, t) \, | \, t \in \mathbb{R}\}=\{u+tv\mid t\in\mathbb{R}\}
$$
where
$u=(1,1,0)$ and $v... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1989447",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Number of different possibilities including repeats so in one of my math classes, we're learning about counting and using this operation called "n choose k" $\binom{n}{k}=\frac{n!}{k!(n-k)!}$, however this operation does not account for the possibilities of repeats, and answers with repeating variables.
An analogy woul... | So for example, if you want to compute the number of ways to choose $3$ toppings out of $7$ not necessarily distinct, and the order in which they are put on the pizza does not matter, here is how you could do it:
*
*Choose three indices $i_1<i_2<i_3$ from $\{1,2,...,9\}$, since $7+3-1=9$.
*Map $i_2\mapsto i_2-1$ an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1989552",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How do I show this without using Euler Formula :$e^{i\frac{\pi}{2}}-i=e^{i\pi}+1$? I would like to show this without using Euler Formula:$$e^{i\frac{\pi}{2}}-i=e^{i\pi}+1$$.?
Note: I multiplied both side by $i$ but i don't succeed .
Edit: I edited the question because i meant Euler formula and the last very related to... | Without defining what $e^z, z \in \mathbb{C}$ is, none of the $e^{i\pi},e^{i\frac{\pi}{2}}$ make sense. And, inevitably, this leads to Euler's formula. Now, depending on the complexity of the course, either
*
*easy - define $e^{i\phi}=\cos{\phi}+i\sin{\phi}$, like this one, page 7, with some reasoning why this defi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1989668",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Let f be an arbitrary, twice differentiable function for which $f''\neq0$ Let f be an arbitrary, twice differentiable function for which $f''\neq0$ . The function $u(x,y)=f(x^2+axy+y^2)$ satifsfies equation $U_{xx}-U_{yy}=0$ then the constant a is.
$f_{xx}(x^2+axy+y^2)(2x+ay)^2+2f_x(x^2+axy+y^2)-f_{yy}(x^2+axy+y^2)(2y... | By the chain rule, $$U_{xx} = 2f'(x^2+axy+y^2) + (2x+ay)^2f''(x^2+axy+y^2)$$ and $$U_{yy} = 2f'(x^2+axy+y^2) + (2y+ax)^2f''(x^2+axy+y^2).$$
Since $U_{xx}-U{yy}=0$, and $f''(x) \neq 0, \forall x \in D_f$, we have $$((2x+ay)^2-(2y+ax)^2)f''(x^2+axy+y^2) = 0 \Rightarrow 4x^2+4axy+a^2y^2 - 4y^2 -4axy - a^2x^2 = 0.$$
In par... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1989883",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Elementary proof for a generalized version of Pascal's hexagon theorem? I saw a generalization version of Pascal's theorem on a book, which is actually proved by deep algebraic geometry theorems (e.g., Cayley-Bacharach-Chasles theorem, or Bézout's theorem). I just wonder whether we can prove it by elementary/synthetic ... | I don't know about synthetic proof but this theorem is the radical axis theorem, seen from a different projective point of view (I.e. Gaged differently). Extend this configuration from the real projective plane to the complex one. Then by applying a complex projective transformation, send the two points $B$ and $E$ to ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1989954",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Solving the following integral $\int_1^c\frac{1}{x} \cos(a_nx)\mathrm{d}x\ $ Solve the following integral
$$\int_1^c\frac{1}{x} \cos(a_nx)\mathrm{d}x$$
Where $$a_{n}=\frac{(2n-1)\pi}{2c}$$
What I tried:
I tried using integration by parts and I got an expression of the form
$$\int_1^c\frac{2}{(a_n)^2x^3} \cos(a_nx)\ma... | As said in comments and answers, the problem and the answer look rather strange.
$$\int \frac{\cos(\alpha x)}x \,dx=\text{Ci}(\alpha x)$$ where appears the cosine integral function which cannot be expressed using any elementary function. $$\int_1^c \frac{\cos(\alpha x)}x \,dx=\text{Ci}(c \alpha )-\text{Ci}(\alpha )\qq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1990054",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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The idea behind the sum of powers of 2 I know that the sum of power of $2$ is $2^{n+1}-1$, and I know the mathematical induction proof. But does anyone know how $2^{n+1}-1$ comes up in the first place.
For example, sum of n numbers is $\frac{n(n+1)}{2}$. The idea is that we replicate the set and put it in a rectangle, ... | Another geometric interpretation.
Start with a line segment, AB. Extend this into a line. Use compassto mark off a point C by setting center at B and passing radius through A. Then lengths, AB=BC. Create point D in a similar way. We have thus constructed a length double that of the original line segment.
This doubling... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1990137",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "56",
"answer_count": 11,
"answer_id": 7
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A Problem in Pigeonhole Principles.
A computer network consists of six computers. Each computer is
directly connected to at least one of the other computers. Show that
there are at least two computers in the network that are directly
connected to the same number of other computers.
I want to solve this problem ... | Note: the most computers any one computer connect to is 5. There just aren't any other computers to connect to.
So you have 5 possible numbers of connections, and 6 computers so... pigeon hole!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1990227",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Simplifying Factorials in Mathematical Induction I'm trying to understand an example for proof by mathematical induction from "Book of Proof (PDF)" (pg 158). I understand the basis step, but I'm not following the simplification in the inductive step, which I've copied below.
Suppose $\sum_{i=0}^{k}i.i!=(k+1)!-1$. Then... | Suppose $\sum_{i=0}^{k}i.i!=(k+1)!-1$. Then:
$\sum_{i=0}^{k+1}i.i!=(\sum_{i=0}^{k}i.i!)+(k+1)(k+1)!$
$((k+1)!-1)+(k+1)(k+1)!$
$(k+1)!+(k+1)(k+1)!-1$
Factor out $(k+1)!$ to get $(1+(k+1))(k+1)!-1$
$(k+2)(k+1)!-1$
Since (k+2)(k+1)! = (k+2), !$(k+2)!-1$
$((k+1)+1)!-1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1990371",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show $x^4+x^2+x+1$ is irreducible in $\mathbb{F}_5[x]$ Show $x^4+x^2+x+1$ is irreducible in $\mathbb{F}_5[x]$.
I need to make use of Eisenstein's criterion, but I'm not sure how.
| Hint: Let $x^4+x^2+x+1=(x^2+ax+c)(x^2+bx+d)$ and you search for $a,b,c,d\in F_5[x]$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1990460",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Pointwise convergence vs norm convergence in $X^*$ Let $X$ be a normed space, $X^*$ its dual space and $\left\|{T}\right\|=\sup\{|T(x)|:\left\|x\right\|=1\}$, the usual norm in $X^*$. Let $(T_n)\subseteq X^*$.
It is true that if $T_n\to 0$ in $X^*$ then $T_n$ converges pointwise to zero, because for every $x\in X$ we h... | When $X $ is infinite-dimensional, the topology of pointwise convergence in $X^*$ can never agree with the topology of norm-convergence. Because the former is weak$^*$-convergence. And in the weak$^*$ topology, the unit ball is compact. In the norm topology, though, the unit ball is compact precisely when $X $ is fini... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1990567",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Closed Convex Set and convex combination with parameter $\frac{1}{2}$ I'm trying to show that given $C$, a closed set:
C is convex $\Leftrightarrow \forall \{x,y\} \subseteq C: \frac{1}{2}x + \frac{1}{2}y \in C$
But i'm stuck at the $\leftarrow$ part. I'm trying to prove by contradiction:
$C$ is closed, $\forall \{x,y\... | Hint: Let $(x,y) \in C^2$, $\tau \in [0,1]$. We want to show that $x + \tau(y-x) \in C$. Let $\epsilon > 0$. Given that $x + j\cdot 2^{-n}(y-x) \in C$ for all $n$ and $j \in \{0,\dots, 2^n\}$ (why ?), we're hoping to find $j_\epsilon$ and $n_\epsilon$ such that:
\begin{align*}
|| x + \tau(y-x) - (x + j_\epsilon \cdot ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1990690",
"timestamp": "2023-03-29T00:00:00",
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Differential equation $f''(x)+2 x f(x)f'(x) = 0$ I am trying to solve,
$ f''(x)+2 x f(x)f'(x) = 0$
with boundary conditions $f(-\infty)=1$ and $f(\infty)=0$. I have found that for instance $f(x) = 3/2 x^{-2}$ but obviously it does not satisfy the proper boundary conditions. Any ideas for a solution?
| Hint:
This belongs to a generalized Emden–Fowler equation.
$f''(x)+2xf(x)f'(x)=0$
$\dfrac{d^2f}{dx^2}=-2xf\dfrac{df}{dx}$
$\therefore\dfrac{d^2x}{df^2}=2fx\left(\dfrac{dx}{df}\right)^2$
Follow the method in http://science.fire.ustc.edu.cn/download/download1/book%5Cmathematics%5CHandbook%20of%20Exact%20Solutions%20for%2... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proof of invertibility of $I+S$ such that $S^* = -S$ We're given a linear transformation $S:V \rightarrow V$ on a finite dimensional inner product space $V$ such that its adjoint is negative $S$ (i.e. $S^* = -S$). We are able to prove that $(v+Sv,v+Sv)=(v,v) + (Sv,Sv)$ and that the kernel of $(I+S) = \{\mathbf{0}\}$.
U... | If $T \colon V \rightarrow V$ if a linear map on a finite dimensional vector space with trivial kernel then $\dim \ker T + \dim \operatorname{Im} (T) = \dim V$ implies that $\dim \operatorname{Im}(T) = \dim V$ and so $T$ is also onto and invertible.
In your case, if you have shown that $I + S$ has trivial kernel it imp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1990943",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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If there is $a>0$ s.t. $f|[0,a]$ and $f|[a, +\infty)$ are uniformly continuous, $f:[0, +\infty) \rightarrow \mathbb{R}$ also is Given $f:[0, +\infty) \rightarrow \mathbb{R}$, suppose that exists $a>0$ such that $f|[0,a]$ and $f|[a, +\infty)$ are uniformly continuous. Prove that $f$ is uniformly continuous.
Any hints on... | The following Lemma is well-known and easy to prove (see, for instance, “General topology” by Ryszard Engelking (Heldermann Verlag, Berlin, 1989)). Let $X$ and $Y$ be topological spaces, $\mathcal F$ be a (locally) finite cover of the space $X$ by its closed subspaces. Given a family $\{f_F:F\to Y | F\in\mathcal F\}$ ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Inclusion exclusion: How many bit strings of length eight do not contain six consecutive $0$'s?
Q. How many bit strings of length eight do not contain six consecutive $0$s?
I solved this problem with my hand.
$$256 - ( 1 + 2 + 2 + 1 + 2 ) = 248$$
I calculated all possible events.
Is my answer right?
And can this prob... | I think your answer is wrong but the idea is good. The number to subtract can indeed be found using the inclusion-exclusion principle, for 3 sets. Can you explain why you subtracted those numbers ?
EDIT: Apologies - I was mistaken. See N. F. Taussig's answer and comment.
| {
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"timestamp": "2023-03-29T00:00:00",
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Cholesky decomposition vs. $\mathbf L\mathbf D\mathbf L^\top$ decomposition In different books and on Wikipedia, you can see frequent mentions of Cholesky decomposition and $\mathbf L\mathbf D\mathbf L^\top$ decomposition is seldom mentioned. Why so? As far as I understand, $\mathbf L\mathbf D\mathbf L^\top$ decomposit... | A frequent reason for preferring $\mathbf L\mathbf D\mathbf L^\top$ over Cholesky is that the latter requires the evaluation of square roots, while the former does not. In symbolics, where one might want to keep the entries radical-free, or in numerics, where the square root operation on a particular machine might be s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1991336",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Finding if two sets are equal I have an formula
$(A\cap C) \cup ( B \cap C) = C - ( A \cap B)$
and i have to prove if this equation is correct.
I transformed it into
$(\forall x\in U: x \in A \wedge x\in C) \vee ( x\in B\wedge x\in C) = x \in C \wedge ( x \notin A \wedge x\notin B) $
how could i further transform t... | Instead of transforming the equation just go after the english meaning: those in A and C or B and C are those in C but not both A and B. This is obvioyusly not true: Consider A=B
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1991442",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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real values of $a$ for which the range of function $ f(x) = \frac{x-1}{1-x^2-a}$ does not contain value from $\left[-1,1\right]$
All real values of $a$ for which the range of function $\displaystyle f(x) = \frac{x-1}{1-x^2-a}$ does not
contain any value from $\left[-1,1\right]$
$\bf{My\; Try::}$ Let $\displaystyle y ... | Given
$$f(x)=\dfrac{x-1}{1-x^2-a}$$
for some $a$
notice that if for $x\ne0$ you multiply both the numerator and denominator by $\dfrac{1}{x}$ you obtain
$$ f(x)=\dfrac{1-\frac{1}{x}}{\frac{1}{x}-x-\frac{a}{x}}\text{ for }x\ne0 $$
Now notice what happens to the value of $f(x)$ for large values of $x$. Regardless of the... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Simple Proof of Obvious Fact. Have you any idea of how to proof the following as simpler as possible?
Let $M$ be family of $n$ finite sets:
$$M = \{X_1, X_2, ..., X_n\}$$
Then
$$\bigr(\forall k = 1, 2, ..., n : |X_{i_{1}}\cup X_{i_{2}}\cup\cdots\cup X_{i_{k}}|\geq k\bigl)\Rightarrow\bigl(\exists \chi_1,\chi_2,...\chi_... | Your obvious fact is a harder part of a generalization of Hall’s Marriage Theorem, proved in 1935 and it probably has no very simple proof (at least I was not told about it in my graph theory lecture course).
| {
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"url": "https://math.stackexchange.com/questions/1991592",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Permutation Groups - Rewrite as disjoint cycles / transpositions Why is this true? "(1243)(3521): Written as a product of disjoint cycles, we get (354)."
Shouldn't (1243)(3521) = (354)(21)? Where did the (21) go?
Thanks!!
| The cycle $(3521)$ sends $3$ to $5$, and the cycle $(1243)$ does nothing to $5$, so the product sets $3$ to $5$. The cycle $(3521)$ sends $5$ to $2$, which $(1243)$ sends to $4$, so the product sends $5$ to $4$. The cycle $(3521)$ leaves $4$ alone, and $(1243)$ sends it to $3$, so the product sends $4$ to $3$. This est... | {
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Prob. 14, Chap. 3, in Baby Rudin: The arithmetic mean of a complex sequence Here's Prob. 14, Chap. 3 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition:
If $\left\{ s_n \right\}$ is a complex sequence, define its arithmetic mean $\sigma_n$ by $$ \sigma_n = \frac{s_0 + s_1 + \cdots + s_n}{n+1... | For (c): Consider $s_{2^m} = m, m = 1,2, \dots, s_n = 0$ for all other $n.$ (We don't have $s_n >0$ for all $n,$ but it will give you the idea.)
| {
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"timestamp": "2023-03-29T00:00:00",
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$f(1+x) = f(x)$ for all real $x$, $f$ is a polynomial and $f(5) = 11$. What is $f\large(\frac{15}{2}\large)$? I was looking through a GRE math subject test practice test (here) and in particular I was confused regarding this question
I chose E because I thought that (since the problem didn't specify) it could be an in... | $$f(x) = a_0 + a_1x^1 + \cdots + a_nx^n$$
$$f(x+1) = a_0 + a_1(x+1)^1 + \cdots + a_n(x+1)^n$$
It can be shown by induction that
$$f(x) = f(x+1) \to a_1=a_2=\cdots=a_n=0$$
Note that
$$f(x) = a_0 + a_1x^1 + \cdots$$
is not a polynomial
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1991889",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 2
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Basis for a topology on $\mathbb{R}$ Let $B = \{[a, b] \mid \forall\, a, b \in \mathbb{R}, a < b\}$. Then $B$ a basis for some topology on $\mathbb{R}$.
Is it true that the set of all closed subsets in $\mathbb{R}$ generate a basis for some topology on $\mathbb{R}$? I know from previous assignments that the lower limit... | Let $X$ be a set and $\mathcal B$ some family of subsets of $X$. Then there exists unique topology with basis $\mathcal B$ if and only if:
$(1)$ $\bigcup\mathcal B = X$
$(2)$ $(\forall B_1,B_2\in\mathcal B)(\forall x\in B_1\cap B_2)(\exists C_x\in\mathcal B)\ x\in C_x\ \wedge\ C\subseteq B_1\cap B_2$
Proof. One directi... | {
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Nilpotent and Idempotent Elements of Ring $\mathbb{Z}_6 \times \mathbb{Z}_2$ I'd just like to make sure I understand nilpotent and idempotent elements of rings properly.
A nilpotent element is an element $b \in R$ s.t. for some $n$, $b^n = 0$. An idempotent element is an element $b \in R$ s.t. $b^2 = b$.
So, for the r... | Yes, you are right and this is quite easy to prove since we have:
$$(a, b)^n=(a^n, b^n)$$
Thus, $(a, b)^n=(0, 0)$ if and only if $a^n=0$ and $b^n=0$, so only pairs of nilpotent elements are nilpotent. Also, if we have two nilpotent elements $a$ and $b$ such that $a^m=0$ and $b^n=0$, then $(a, b)^{\max(m, n)}=0$ and thu... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Relation between left null space, row space and cokernel, coimage I know the null space and the column space of a matrix $A$ can be identified with the kernel and the image of the associated homomorphism $L_A: x \mapsto Ax$, once a basis is chosen.
I was reading this Wikipedia article that calls the left null space and... | Suppose the matrix $A$ is $m\times n$ and has rank $k$.
Thus $A$ defines a linear map $L_A\colon \mathbb{R}^n\to \mathbb{R}^m$. The null space has dimension $n-k$ and the image has dimension $k$.
The cokernel $\mathbb{R}^{m}/\operatorname{im}L_A$ has dimension $m-k$. The coimage $\mathbb{R}^n/\ker L_A$ has dimension $n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1992261",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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$f(x)=x/1$ is not injective? In chapter 3 of Atiyah and Macdonalds Intro to Comm Algebra there is the statement
"We also have a ring homomorphism $f:A\rightarrow S^{-1}A$ defined by $f(x)=x/1$. This is not in general injective." (Note: $A$ denotes an integral domain and $S$ a multiplicatively closed subset of $A$.)
How... | EDIT: The answer below the line assumes $A$ is just a ring. In case $A$ is an integral domain, that answer of course does not apply, and your reasoning is right . . .
. . . assuming $S$ doesn't contain $0$. If $0\in S$, then $S^{-1}A$ is the trivial ring, and $f$ is extremely not injective.
The localization is a bit ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1992449",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Are partitial differential operators commutative? I am trying to convert $\Delta=\nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}$ to polar coordinates. If anyone has any references on how to do that I would appreciate it.
In my evaluation, I am messing with a lot of partials. Alghough, I am... | What you're looking for is the "Polar Laplacian". Googling "Polar Laplacian Derivation" leads to many papers such as this one that go through step-by-step with how to derive it.
If after reading this you're still interested in a challenge, try deriving the Laplacian for Spherical or Cylindrical coordinates.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1992540",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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How do you "simplify" the sigma sign when it is raised to a power? How do you simplify the following expression:
$$\left(\sum^{n}_{k=1}k \right)^2$$
I am supposed to show that
$$\left(\sum^{n}_{k=1}k \right)^2 = \sum^{n}_{k=1}k^{3} $$
The problem is I do not really know how to manipulate the sigma sign. I know that I ... | $$\begin{align}
\sum^{n}_{k=1}k&=\frac{n(n+1)}{2}\\
\left(\sum^{n}_{k=1}k\right)^2&=\left(\frac{n(n+1)}{2}\right)^2\\
\left(\frac{n(n+1)}{2}\right)^2&=\frac{n^2(n+1)^2}{4}\\
\sum^{n}_{k=1}k^3&=\frac{n^2(n+1)^2}{4}\\
\therefore \left(\sum^{n}_{k=1}k\right)^2&=\sum^{n}_{k=1}k^3
\end{align} $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1992625",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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How many #8 digits can you build?
How many different 8-digit numbers can be formed using two 1s, two 2s, two 3s, and two 4s such that no two
adjacent digits are the same?
So we have $0 -> 9 = 10$ options for the digits but with the constraints we have something different.
Cases:
$1$ _ _
$2$ _ _
$3$ _ _
$4$ _ _
W... | Use Inclusion-exclusion principle.
There are $\frac{8!}{2^4}$ 8-digits numbers that can be formed using two 1s, two 2s, two 3s, and two 4s.
i) How many of these 8-digits numbers have two adjacent 1s?
There are $7\cdot \frac{6!}{2^3}$ such numbers.
ii) How many of these 8-digits numbers have two adjacent 1s and two ad... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Analytical solution of quadratic programming when $H$ is not p.d.? Considering the unconstrained quadratic programming: $$\min_x \frac{1}{2}x^\top Hx+c^\top x$$
if $H$ is not positive definite, can we still calculate the analytical solution?
| We consider the function
$$ f \colon \def\R{\mathbf R}\R^n \to \R, \quad x \mapsto \frac 12 x^tHx + c^t x $$
for $H\in \def\M{\mathrm{Mat}}\M_n(\R)$, $c \in \R^n$. Taking derivatives gives
\begin{align*}
f'(x)h &= \frac 12 x^t(H+H^t)h + c^t h\\
f''(x)[h,k] &= \frac 12 h^t(H+H^t)k
\end{align*}
So we see that the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1992986",
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"source": "stackexchange",
"question_score": "3",
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First-order logic and NBG I know that NBG set theory can be viewed as an interpretation of the first-order logic with equality. However, I do not really understand how do I get this interpretation.
As I see it:
I call variables "classes" and add the class membership relation $\in$. I call a class $x$ a "set" if $\exi... | It's not that NBG is an interpretation of FOL, it's that NBG is a theory in FOL, in the same way that e.g. the theory of rings is a theory in FOL.
The language of NBG has a single binary relation, "$\in$," and no constants or functions. So there's no need to interpret those (any more than we would demand that a ring u... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1993104",
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"source": "stackexchange",
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Inverses of bijective functions. Whenever I see a $1-1$ and onto function from $\mathbb{R}$ to $\mathbb{R}$,I wonder how do I even go about finding the inverse.
Is there a general way of finding the inverse of a bijective function $f :\mathbb{R}\to\mathbb{R}$? Or is it really possible that inverse of all bijective f... | There are an uncountable number of bijections on $\mathbb{R}$, and only a countable set of "formulas" consisting of finite strings of mathematical symbols from some (finite) alphabet. Therefore, by the pigeonhole principle, there are bijections on $\mathbb{R}$ where no formula (with any set of finite mathematical symbo... | {
"language": "en",
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Can`t find the integral $\int\frac{dx}{\sqrt x\sin(x)}$ $\int\frac{dx}{\sqrt x\sin(x)}$ Im stuck in this problem for about 40 minutes. Any clue how to solve this?
The actual problem is an ODE problem, it is given $x^2y''+xy'+(x^2-1/4)y=0$ and one of the solutions is $y_1(x)=\sin(x)/\sqrt(x)$. So I need to find $v=\int{... | The solution you have suggests that you let
$$
y(x)=v(x)/\sqrt{x}.
$$
Doing so, your differential equation will turn into
$$
x^{3/2}(v''(x)+v(x))=0.
$$
I'm sure you can take it from there.
(I don't see where you got your integral from, so I cannot help you with that.)
| {
"language": "en",
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If $M$ is a local martingale and $τ_k^n$ is a stopping time, then $t↦\sum_{k∈ℕ}M_{τ_{k-1}^n}\left(M_{τ_k^n∧t}-M_{τ_{k-1}^n∧t}\right)$ is a martingale Let
*
*$T>0$
*$(\Omega,\mathcal A,\operatorname P)$ be a probability space
*$(\mathcal F_t)_{t\in[0,\:T]}$ be a filtration on $(\Omega,\mathcal A)$
*$M$ be an almos... | Show that each term in the sum is a martingale. To wit, a stopped local martingale is a local martingale, and since $M_{\tau_k^n\wedge t}$ is a.s. bounded (by $k2^{-n}$), each $M_{\tau_k^n\wedge t}$ is a martingale, as is the difference $M_{\tau_k^n\wedge t}-M_{\tau_{k-1}^n\wedge t}$. As this difference vanishes on $... | {
"language": "en",
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"source": "stackexchange",
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What is mathematical logic? What does mathematical logic mean?
In the book Analysis 1 by Terence Tao, it says:
The purpose of this appendix is to give a quick introduction to
mathematical logic, which is the language one uses to conduct
rigourous mathematical proofs.
Checking Wikipedia:
Mathematical logic is oft... | Mathematical logic is a strange beast.
It is a perfectly ordinary branch of mathematics whose goal is ... to study mathematics itself.
Thus, the different branches of mathematical logic are devoted to the study of some basic building blocks of mathematical practice : language, model, proof, computation.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1993596",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 6,
"answer_id": 0
} |
Compute the limit for $x \to \infty$ of a certain rational function Compute the following limit:
$$
\lim_{x \to \infty}
\frac{9x^8 + 5x^2 − 6}{
3x^8 + 2x^4 + 1}
$$
The answer may be
A. 2
B. 3
C. 5
D. 7
| Hint: Divide numerator and denominator by $x^8$ to get:
$$\lim_{x \to \infty}\frac{9+5/x^{6}-6/x^8}{3+2/x^4+1/x^8}$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Transitive subgroup of symmetric group I'm working on the following question, and honestly have no idea how to begin. Any hints would be greatly appreciated!
Let $H$ be a subgroup of $S_n$, the symmetry group of the set $\{1,2,\dots, n\}$. Show that if $H$ is transitive and if $H$ is generated by some set of transposi... | Proof by induction on $n$. Case $n\le 2$ is trivial.
Let $X$ be the set of transpositions in $H$, $X_1$ the set of transpositions in $X$ fixing 1 and $K$ the subgroup generated by $X_1$.
We have to prove that $K$ is transitive on $\{2,\dots,n\}$. If true, induction hypothesis implies $K$ is the full symmetry group on ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1993832",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
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Arithmetic or Geometric sequence? Given a sequence:
$$1, \frac12, \frac13, \frac14, \frac15,...$$
Its explicit formula can be given as:
$a(n) = \frac1n$ where $n \ge 1$.
I actually want to know is it a geometric sequence or an arithmetic one?
I tried finding common ratio and the common difference in this sequ... | This is not a geometric series.
$a_1=1, a_2=\frac12, a_3=\frac13$
If this is a geometric sequence, then it is necessary that $\frac{a_2}{a_1}=\frac{a_3}{a_2}$
$$\frac{a_2}{a_1}=\frac{1}{2}$$
$$\frac{a_3}{a_2}=\frac{2}{3}$$
The two numbers are different, hence it is not a geometric sequence.
Similarly, you can verify th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1993989",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Repeated Differences of the series 1^n 2^n 3^n 4^n… I kind of stumbled across this pattern recently and cant make any sense of it. Attached is my work explaining what I did. If there is already an answer to this please post the link. Thanks!
| It's more natural to start the sequence at $0$: for example,
$$0 \quad 1 \quad 4 \quad 9$$ $$1 \quad 3 \quad 5$$ $$2 \quad 2$$ $$0$$
from which you can read off the leftmost side that the top (the sequence $n^2$, $n \in \mathbb{N}$) is actually $$\color{red}{0} \cdot \binom{n}{0} + \color{red}{1} \cdot \binom{n}{1} + \... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Does globally stability implies existence of Lyapunov function? I recently had a chance to ask a professor this question and he mentioned that this is true for ordinary differential equations (but not for delay differential equation). He did not remember the reference, I could not find a reference for this, so can anyo... | You will find what you need in this paper and the references of Karafyllis:
Converse Lyapunov–Krasovskii theorems for systems described by neutral functional differential equations in Hale's form
Pierdomenico Pepe & Iasson Karafyllis
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Pointwise convergence of $X_n$ vs $X_nI_{\{|X_n|\leq c_n\}}$ and of $\sum X_n$ vs $\sum X_nI_{\{|X_n|\leq c_n\}}$
Let $\{X_n,n\geq 1\}$ be a sequence of random variables, and $\{c_n,n\geq1\}$ a positive sequence. Let also $\sum_n P(|X_n|> c_n)<\infty$. Prove:
*
*If $Y_n=X_nI_{\{|X_n|\leq c_n\}}$ and $P\left( ... | By Hypothesis, $\sum_n P(\{X_n\neq Y_n\})<\infty$.
By Borel-Cantelli Lemma, $P\left( {\mathop {\lim \inf }\limits_n {\mkern 1mu} {{\{X_n=Y_n\}}}} \right) = 1.$
Put $A={\mathop {\lim \inf }\limits_n {\mkern 1mu} {{\{X_n=Y_n\}}}}$
For every $\omega \in A, X_n(\omega)=Y_n(\omega)$ for all large n.
This shows, for ever... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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double integral wrt minimum How can one calculate the following integral?
$$
\int_0^1{\int_0^1 xy \quad d[min(x,y)]}
$$
I have no idea how to hande the $d(min(x,y))$.
Does anyone have an idea for this problem?
| Certainly this is a (Lebesgue-)Stieltjes integral. Nevertheless, fortunately $C_1=xy$ and $C_2=\min(x,y)$ are copulas and for copulas the following holds (partial dderivatives exist a.e. of course):
$$\iint_{[0,1]^2}C_1(x,y)\,\mathrm{d}C_2(x,y)=\frac{1}{2}-\iint_{[0,1]^2}\frac{\partial}{\partial x}C_1(x,y)\frac{\partia... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1994383",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Let $p$ be an odd prime. Suppose that $a$ is an odd integer and also $a$ is a primitive root mod $p$. Show that $a$ is also a primitive root mod $2p$. Let $p$ be an odd prime. Suppose that $a$ is an odd integer and also
$a$ is a primitive root modulo $p$. Show that a is also a primitive root modulo
$2p$.
Any hints will... | A way is like this (you may have to justify the points a bit):
*
*There are $p-1$ invertible classes mod $2p$, the same number as for $p$.
*$a$ is invertible mod $2p$.
*The multiplicative order of $a$ mod $p$ is $p-1$.
*The multiplicative order of $a$ mod $2p$ is at least as large as the one mod $p$.
From thi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1994510",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that the restriction $f\vert_A$ is continuous Let $S\subset \mathbb{R}$ and $A\subset S$. Let $f:S\rightarrow \mathbb{R}$ be a continuous function. Then the restriction $f\vert_A$ is continuous.
| By definition of the subspace topology, if $(X,\tau)$ is a topological space then $\{U\cap A | U\in\tau\}$ are the open sets of $A\subseteq X$. But then if $O$ is an open subset of $\Bbb R$ we have that
$$f\big|_A^{-1}(O) = f^{-1}(O)\cap A$$
but since $f$ is continuous, this is the intersection of open sets, hence is i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1994627",
"timestamp": "2023-03-29T00:00:00",
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Describe the set whose points satisfy the following relation: $|z^2 - 1| < 1$ There is a hint which states to use polar coordinates, but I feel like that complicates the problem more. As far as trying it myself, I get lost very early on. If we take $z = r(\cos{\theta} + i\sin{\theta})$, then we have
$|r^2(\cos{2\theta}... | If you use $re^{i\theta}$ as the polar representation than you get $$|r^2e^{2i\theta}-1|<1\Rightarrow
|(re^{i\theta}+1)(re^{i\theta}-1)|<1$$ which might be enlightening....
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1994750",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Compact subset of the space of all bounded sequences of real numbers Let $X$ be the metric space of bounded sequences of real numbers $ x = (x_n) $ with the metric $ d(x,y) = \sup_n |x_n - y_n| $. Show that the set
$$ Y = \{ x = (x_n) \in X \mid |x_n| \leq c_n = \text{ const for all } n \} $$
is compact in $X$ if an... | We can show total bounded in the following way: let $\varepsilon$ be a positive number. Choose $n_0$ such that if $n\geqslant n_0$, then $\left|c_n\right|\lt\varepsilon$. The set $S=\prod_{j=1}^{n_0-1} [-c_j,c_j]$ is compact for the metric $d_{n_0}(x,y) :=\max_{1\leqslant j\leqslant n_0-1}\left|x_j-y_j\right|$. Choose... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to prove the sum of deviations from the mean is minimum? I am trying to prove that the sum of deviations from the mean is minimum. I saw this prove in a book
where $\bar{X}$ is the mean and $Y\neq \bar{X}$. What I am confused about is where does the zero in the last line come from?
| The middle line should in fact be
$\displaystyle \sum_{i=1}^n (X_i-Y)^2 = \sum_{i=1}^n (X_i-\overline{X})^2 + 2(\overline{X}-Y)\sum_{i=1}^n (X_i-\overline{X}) + \sum_{i=1}^n (\overline{X}-Y)^2$
and you then have $\displaystyle\sum_{i=1}^n (X_i-\overline{X})=\left(\sum_{i=1}^n X_i\right) - n \left(\frac1n\sum_{i=1}^n X... | {
"language": "en",
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Do the limits exist? Determine whether the following limits exist and determine them in case of convergence:
1.)
$$
\lim_{n\to\infty}\frac{\arctan(\exp(-n))}{\exp(n^2)}-\frac{\ln(n+1)}{n}.
$$
2.)
$$
\lim_{x\downarrow 0}\frac{\tan(x^2)-\ln(\ln(x+1))}{\ln(1/x)}
$$
Here's what I tried.
1.) Considering the first summand,... | You are right about the first limit, with a slight note that $\arctan(\exp(-n))$ does not tend to $\pi/2$ as $n \to +\infty$ but rather to zero. I think you missed the minus in the exponent.
In the second limit, yes, you can use a second iteration of l'Hôpital's rule. That's a pretty common situation. One just needs to... | {
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"url": "https://math.stackexchange.com/questions/1995116",
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"source": "stackexchange",
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Closest point on a plane to a surface. And vice versa. Find the point on $z=1-2x^2-y^2$ closest to $2x+3y+z=12$ using Lagrange multipliers.
Point on surface closest to a plane using Lagrange multipliers
Although the methods used in the answers are helpful and do work, my professor told me that the way he wants us to do... | Answering to "I don' understand..claim that the distance will be minimized when the normals are parallel".
On a pure intuitive basis (not pretending to be rigorous) consider a point on the given surface, and the tangent plane to the surface at that point.
In a small domain around the point, the surface is "approximated... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1995257",
"timestamp": "2023-03-29T00:00:00",
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Prove using Induction | Tricky one! I actually need some help. I want to prove using simple induction that
Q.1) $2^n > n^3$ for all $n \geq 10 $
I tried solving it like this...
Base Step:
$n = 10$:
$2^{10} > 10^3 = 1024 > 1000$ So, that's true and fine.
Inductive step:
Suppose $2^n > n^3$ is true for some ... | Hint:
From the induction hypothesis, you deduce that
$$2^{n+1}=2\cdot 2^n>2n^3,$$
hence by transitivity, it's enough to show that $2n^3\ge (n+1)^3$, or $\Bigl(1+\dfrac 1n\Bigr)^3\le2$.
Observe that
$$\Bigl(1+\dfrac 1n\Bigr)^3=1+\frac3n+\frac3{n^2}+\frac1{n^3}\le 1+\frac9n\quad\text{(why?)}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1995387",
"timestamp": "2023-03-29T00:00:00",
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} |
If $\lambda = i$ is an eigenvalue of $A \in \mathbb R^{n \times n}$, explain why the rank of $A^3 + A$ is less than $n$
$A$ is a $n \times n$ matrix with real entries and $\lambda = i$ is an eigenvalue of $A$. Explain why the rank of $A^3 + A$ is less than $n$.
I understand completely how to get eigenvalues, as well ... | Let $\vec{x}$ be the eigenvector corresponding to the eigenvalue $\lambda=i$. By definition, this means that $A\vec{x}=\lambda\vec{x}=i\vec{x}$.
Consider the product $A^3\vec{x}$ and note that we can move the scalars $i$ from each product to the front, yielding $A^3\vec{x}=A(A(A\vec{x}))=iA(A\vec{x})=i^2A\vec{x}=i^3\ve... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1995525",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Linearly independent set of vectors in a normed space
Lemma:
Let $\left \{ v_{1},\cdot \cdot \cdot ,v_{n} \right \}$ be a linearly independent set of vectors in a normed space $V,\left \| \cdot \right \|$. Then, there exists a constant $c>0$ such that for any scalars $\alpha_{1},\cdot \cdot \cdot ,\alpha_{n}$:
$\lef... | $$||\alpha^1v_1+...+\alpha^nv_n||\le ||\alpha^1v_1||+...+||\alpha^nv_n||=|\alpha^1| ||v_1||+...+|\alpha^n| ||v_n||\le max(||v_i||)(|\alpha^1|+...|\alpha^n|)$$
Any $c\ge max(||v_i||)$ is ok
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1995656",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Suppose $t>0$ and $0<\delta<1$. Prove: $\int_{0}^{t} (t-s)^{-1/2}s^{-1+\delta/2}ds=t^{(-1+\delta)/2}$. Suppose $t>0$ and $0<\delta<1$. Prove: $\int_{0}^{t} (t-s)^{-1/2}s^{-1+\delta/2}ds=t^{(-1+\delta)/2}$.
I tried dilation but it didn't work. I appreciate if anyone can give some hints. Thank you.
| Using the same steps as DonAntonio in his answer,$$\int_0 ^t \frac{s^{\frac{\delta }{2}-1}}{\sqrt{t-s}}\,ds=\sqrt{\pi }\frac{\Gamma \left(\frac{\delta }{2}\right) }{\Gamma \left(\frac{\delta +1}{2}\right)}t^{\frac{\delta
-1}{2}}$$ If $\delta$ is close to $0$, Taylor series give $$\sqrt{\pi }\frac{\Gamma \left(\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1995773",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Calculate exponential limit involving trigonometric functions Calculate the following limit:
$$\lim_{x \rightarrow 0} \left( \frac{\tan x}{x} \right) ^ \frac{1}{\sin^2 x}$$
I know the result must be $\sqrt[3]{e}$ but I don't know how to get it. I've tried rewriting the limit as follows:
$$\lim_{x \rightarrow 0} e ^ {\l... | You have an indeterminate form of kind $1^{\infty}.$ You can solve this type of problems via:
$$\lim_{x\to a} f(x)^{g(x)}=e^{\lim_{x\to a} (f(x)-1)g(x)}.$$
Edit
If $\lim_{x\to a}f(x)=1$ then it is $$\lim_{x\to a} f(x)^{g(x)}=\lim_{x\to a} (1+f(x)-1)^{\dfrac{f(x)-1}{f(x-1)}g(x)}=\lim_{x\to a} \left( (1+f(x)-1)^{\dfrac{1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1995940",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
When is $E[f(X)]=0$ for even functions Let $X$ be standard normal and $f$ a function that satisfies
*
*$f(0)=0$
*$f$ is even
*$(x-{\rm sign}(x) \cdot a) \cdot x \le f(x) \le (x+{\rm sign}(x) \cdot a) \cdot x$, for all $x$, and some fixed $a>0$. Moreover, these bounds are asymptotcily tight.
I am either trying... | If $a < \sqrt{\dfrac{\pi}{2}}$, then for any $f$ satisfying the given conditions, we have:
$\mathbb{E}[f(X)]$ $= \displaystyle\int_{-\infty}^{\infty}\dfrac{1}{\sqrt{2\pi}}e^{-x^2/2}f(x)\,dx$ $= \displaystyle\int_{0}^{\infty}\sqrt{\dfrac{2}{\pi}}e^{-x^2/2}f(x)\,dx$
$\ge\displaystyle\int_{0}^{\infty}\sqrt{\dfrac{2}{\pi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1996165",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Trace of product of three Pauli matrices Consider the four $2\times 2$ matrices $\{\sigma_\mu\}$, with $\mu = 0,1,2,3$, which are defined as follows
$$
\sigma_0 =\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right)
$$
$$
\sigma_1 =\left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right)
$$
$$
\sigma_2 =\left( \... | First I note a few things about the defined matrices:
1) Commutation: $[\sigma_\mu,\sigma_\nu] = 2 i \epsilon_{0 \mu \nu \rho} \sigma_\rho$
2) Anti-commutation: $\{ \sigma_\mu ,\sigma_\nu \} = 2 \delta_{\{\mu \nu} \sigma_{0\}} - 4 \delta_{\mu 0} \delta_{\nu 0} \sigma_0$
Where $a_{ \{b c} d_{i\}} = a_{bc} d_i + a_{ib} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1996297",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Best of seven with different home/away win rates? I'm having some issues with a question from my intro probability course and I'm hoping you can help. Here's the question:
In the World Series in baseball and in the playoffs in the National Basketball Association and the National Hockey Association, the winner is determ... | The ordering of the games doesn't matter with this stripped down model (with no "momentum" factor, say).
To see, this, note that we can always assume that all seven games are played. Even though the series is probably decided before game $7$, playing the series out correctly determines the winner by simple majority. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1996544",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Proof that $\prod_{k=1}^{n-1}(1+\frac1k)^k = \frac{n^n}{n!}$ for all $n \in \Bbb N \ge 2$ I've tried to prove this for a while now, but I can't get it:
$\prod_{k=1}^{n-1}(1+\frac1k)^k = \frac{n^n}{n!}$ for all $n \in \Bbb N \ge 2$
Solution:
$\prod_{k=1}^{(n+1)-1}(1+\frac1k)^k=\frac{(n+1)^{(n+1)}}{(n+1)!}$
$\left(1+\fra... | The following telescopic product in disguise:
$$ \prod_{k=1}^{n-1}\left(1+\frac{1}{k}\right) = n \tag{1}$$
leads to
$$ \prod_{k=1}^{n-1}\left(1+\frac{1}{k}\right)^k = \frac{n^n}{\prod_{k=1}^{n-1}\left(1+\frac{1}{k}\right)^{n-k}}=\frac{n^n}{n\cdot(n-1)\cdot\ldots\cdot 1}=\frac{n^n}{n!}.\tag{2}$$
In the opposite directio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1996735",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Why $F_2[X]/(X^2+X+1)$ has $4$ elements and what are those? I don't understand the three claims that some $F_a[X]/(p(x))$ has some $n$ elements in the following text (from Adkins' Algebra):
For example Why $F_2[X]/(X^2+X+1)$ has $4$ elements and what are those? I think that since $(X^2+X+1)$ is an ideal so $F_2[X]={\{... | Hint: What $R=\mathbb{F}_2/(X^2+X+1)$ means that $X^2+X+1=0$ in $R$. Therefore, any time that you see a power of $X$, you can reduce it to a lower power using $X^2=-X-1=X+1$ (we can drop the negatives since $-1=1$ in characteristic $2$).
For example,
$$
X^3\equiv X(X+1)=X^2+X\equiv (X+1)+X=2X+1=1.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1997022",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
$G$ a connected topological group, U open, then $\mathcal{U} = \cup_{i \in \mathbb{N}}\; U^i$ is a subgroup of $G$. Let $G$ a connected topological group, and $U$ an open subset of $G$ that contains the identity element of $G$. I want to solve the following problem:
If $U^n$ is the set of the products $u_1 u_2 ...u_n... | Choose an open set $V\subseteq U$ containing the identity such that if $g\in V$ then $g^{-1}\in V$ (for instance, $V=U\cap U^{-1}$). As you say, you can then easily show that $\mathcal{V}=\bigcup_{i\in\mathbb{N}} V^i$ is an open subgroup of $G$. Thus $\mathcal{V}=G$. But $\mathcal{V}\subseteq\mathcal{U}$ since $V\su... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1997162",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Why does a unitary in the Calkin algebra always lift to an (co-)isometry? Cf. the the title, consider a separable infinite-dimensional Hilbert space H, and the short exact sequence $$0 \to \mathcal{K}(H) \to \mathcal{B}(H) \to \mathcal{Q}(H) \to 0,$$where $\mathcal{K}(H)$ is the compact operators and $\mathcal{Q}(H)$ i... | If $u \in Q(H)$ is unitary, let $T_1 \in B(H)$ such that $\pi(T_1) = u$, then consider
$$
T = T_1R^{-n}
$$
where $R$ denotes the left or right shift (which exists on your separable infinite dimensional Hilbert space) so that $T$ has index zero. Now $\exists S_1 \in GL(B(H))$ such that
$$
T-S_1 \in K(H)
$$
Hence, $S:=S_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1997315",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
If $A,B,$ and $C$ are sets, then $A\times(B-C)$ = $(A \times B)$ $-$ $(A \times C)$. If $A,B,$ and $C$ are sets, then $A\times(B-C)$ = $(A \times B)$ $-$ $(A \times C)$.
Proof. Observe the following sequence of equalities.
$$\begin{align}
A\times(B-C) &= \{(x,y)\} : (x \in A) \wedge (y \in (B-C))\} \, (\text{Definition... | First, when you make a proof with logic, don't use the equality symbol, we use the implication symbol because the implication symbol is defined for the logic propositions.
So, i'm gona show first that $(A \times B) -(A\times C) \subseteq A \times (B-C)$
Let $(x,y)\in ((A \times B) -(A\times C))
\implies (x,y)\in (A \t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1997439",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Interesting Topic for Proofs Class I am teaching one of those university level classes where students learn enough proof theory to do their higher undergraduate courses. In such a class, you need things for the students to prove. What seems to be the norm in textbooks is a little bit of number theory (divisibility, o... | Bezier curves (quadratic, and then cubic ; no need to go further) and splines curves provide a wealth of interesting subjects. The associated proofs are often short and rarely difficult and mix geometry (barycenters, 2D or 3D), analysis (parameterized curves), linear algebra (systems solving, matrix exponential, descri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1997557",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Calculate this limit : $\lim_{x\rightarrow +\infty}\left[x\left(4\arctan\left(\frac{x+1}{x}\right)-\pi\right)\right]$ Calculate the limit
$$\lim_{x\rightarrow +\infty}\left[x\left(4\arctan\left(\frac{x+1}{x}\right)-\pi\right)\right]$$
Neither L'Hospital's rule nor Taylor expansions are allowed,
| Herein, we present an approach that relies on only (1) a set of inequalities for the arctangent function, obtained using only elementary geometry, and (2) the squeeze theorem. To that end, we begin with the following primer.
PRIMER:
I showed in THIS ANSWER, using only elementary inequalities from geometry, that the a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1997709",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 0
} |
Calculating range of parameters for parametrization of hemisphere. I have a hemisphere $x^2+y^2+z^2=1\ ;z\geqslant 0.$ I want to represent it by the vector representation $$\vec r(u,v)=\sin u \cos v\ \hat i+\sin u\sin v \ \hat j+\cos u \ \hat k$$
I am having a very silly problem of figuring out the ranges of $u\ \&\ v$... | Correction: Take $u\in[0, \pi/2]$ then $v$ is simply $0$ to $2\pi$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1997801",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Proof that field trace in $GF(2^k)$ maps half of the elements to 0 and the other half to 1. I'm reading a proof on the solutions of equation $a = z^z + z$ in $GF(2^k)$, that is, the finite field with $2^k$. At some point it uses that the trace function defined as $Tr(a) = a + a^2 + \cdots + a^{2^{k-1}}$ maps half of t... | The trace is a linear form, so it is a map
$$Tr: GF(2^k)\to GF(2).$$
But then linear algebra tells us (rank-nullity) that the null space of this surjective (see below) map is isomorphic to $GF(2^{k-1})$, because it is a vector subspace of $GF(2^k)$ of dimension $k-1$. By definition the null space is things that map t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1997975",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Pumping lemma to prove the set of all words such that the sum of the number of 0s and 1s occurring in it is an even number is not regular In the alphabet {0, 1, 2}, how can I prove using the pumping lemma that there is not a regular expression that can describe the set of all words such that the sum of the number of 0s... | You can’t, because this language is regular. The easiest way to see this is to design a finite state automaton $M$ that recognizes the language. $M$ needs only two states, $q_0$ and $q_1$; $q_0$ will be the initial state and the only acceptor state. $M$ should change state whenever it reads a $0$ or a $1$, and it shoul... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1998128",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Damped pendulum equation Given the equation of a damped pendulum:
$$\frac{d^2\theta}{dt^2}+\frac{1}{2}\left(\frac{d\theta}{dt}\right)^2+\sin\theta=0$$
with the pendulum starting with $0$ velocity, apparently we can derive:
$$\frac{dt}{d\theta}=\frac{1}{\sqrt{\sqrt2\left[\cos\left(\frac{\pi}{4}+\theta\right)-e^{-(\theta... | HINT...Write $$\dot{\theta}=x$$
Then you have $$x\frac{dx}{d\theta}+\frac 12x^2+\sin\theta=0$$
This is a Bernoulli Differential Equation
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1998244",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Potential Energy of a string (Wave Equation PDEs) This may turn out to be an elementary question, but I'm having trouble understanding where this actually comes from.
In my PDEs course, we're learning about the wave equation. We're given the following problem on our current homework set.
Consider a taught string $0 ≤ ... | Just use the approximation
$$
(1 + x)^p \approx 1 + px
$$
for $x \ll 1$. From this
$$
\sqrt{dx^2 + dy^2} - dx = dx\sqrt{1 + \left(\frac{\partial y}{\partial x} \right)^2} - dx \approx dx\left[ 1 + \frac{1}{2}\left(\frac{\partial y}{\partial x} \right)^2\right] - dx = \frac{1}{2}\left(\frac{\partial y}{\partial x} \righ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1998367",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Is the truth value of $(F \iff T) \iff (T \iff F)$ True? I think $(F \iff T) \iff (T \iff F)$ is true, but would like some confirmation.
| Yes, it is indeed true. Remember that "$A\iff B$" is true if and only if $A$ and $B$ have the same truth value - both true, or both false. So both $F\iff T$ and $T\iff F$ are false, and so in turn "$(F\iff T)\iff (T\iff F)$" is true.
Here's a fun fact (you can check it for a few small cases, and then prove it by induct... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1998527",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
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