Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Find $\lim\limits_{x\to\frac\pi2}(1-\sin x)\tan^2 x.$ I'm stuck finding following limit.
$$\lim_{x\to\frac\pi2}(1-\sin x)\tan^2 x.$$
Attempt: I tried to use L'Hopital rule but I cant find the solution
| I usually advise to “go at $0$”: do the substitution $x=\pi/2-t$, so the limit becomes
$$
\lim_{t\to0}(1-\cos t)\cot^2t=
\lim_{t\to0}\frac{1-\cos t}{\sin^2t}\cos^2t=
\lim_{t\to0}\frac{1-\cos t}{t^2}\frac{t^2}{\sin^2t}\cos^2t
$$
If you want to use l'Hôpital, do
$$
\lim_{x\to\pi/2}\frac{1-\sin x}{\cot^2x}
\overset{\mathr... | {
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"url": "https://math.stackexchange.com/questions/1526126",
"timestamp": "2023-03-29T00:00:00",
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Describe the ring $\mathbb{Z}[x]/(x^2)$.
Describe the ring $\mathbb{Z}[x]/(x^2)$.
Attempt: $\mathbb{Z}[x]/(x^2)$ can be thought as the linear polynomials with integer coefficients.
So $\mathbb{Z}[x]/(x^2) = \{a + bx + (x^2) : a,b \in \mathbb{Z}\}$.
What are other things I could describe?
Thank you!
| This is actually a very interesting ring:
You can consider 'x' an "infinitesimal" in a sense.
Let $A = a + x a'$ and $B = a + x a'$ then we have
*
*$A + B = (a+b) + x(a'+b')$
*$AB = (ab) + x(a'b + ab')$
The first component is the normal operation, but the second component follows the same formulas as differentiat... | {
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"url": "https://math.stackexchange.com/questions/1526221",
"timestamp": "2023-03-29T00:00:00",
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Direct Product and Direct sum of a Category (Integer divide)
For the following category, give a definition of the direct product and direct sum; determine if they exist; if they do, give an explicit description, and give an explicit description of the direct product and the direct sum over an empty set:
The category ... | (Direct sum) In the category $D\Bbb Z$, for $a,b\in \Bbb Z$,
the direct sum is
$$
a \coprod b := \operatorname{lcm}(a,b),
$$
the least common multiple of $a,b$ (which we'll take to be $\ge 0$).
Clearly, $a\!\mid\!\operatorname{lcm}(a,b)$ and $b\!\mid\!\operatorname{lcm}(a,b)$.
That is, there are morphisms
$$
a\longr... | {
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Nonhomgeneous Linear Differential Equation: Harmonic Oscillator
Consider frictionless harmonic oscillator (w/ m = 1) driven by an external force $f(t) = A\sin{\omega t} $, so that $$\frac{d^2 x}{dt^2} + \omega_0^2x = A\sin{\omega t}. $$ Show that the particular solution for $\omega \neq \omega_0 $ is $$x_p(t) = \frac{... | Hint:
Assume the particular solution $$y_p=C_1\cos w t+C_2\sin wt$$
$$x_p'=-wC_1\sin w t+wC_2\cos w t$$
$$x_p''=-w^2C_1\cos w t-w^2C_2\sin w t$$
then substitute them to get
$$-w^2C_1\cos w t-w^2C_2\sin w t+w_0^2(C_1\cos w t+C_2\sin wt)=A\sin w t$$
$$(-w^2C_1+w_0^2C_1)\cos w t+(-w^2C_2+w_0^2C_2)\sin w t=0\cos w t+A\si... | {
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Rewrite exponential in form $a+bi$ My professor lately has become really bad at giving examples and any help with an example would be great.
if I have $e^{(2+3i)}$ how do I rewrite it as $a+bi$? is it just $2+3i$ in this case?
| No, it is not simply $2+3i$. Use Euler's formula, i.e.
$$e^{ix} = \cos x + i \sin x$$
so that
$$e^{a+bi} = e^ae^{bi} = e^a(\cos b + i \sin b).$$
| {
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"timestamp": "2023-03-29T00:00:00",
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Find an integer n such that $U(n)$ is isomorphic to $\Bbb Z_2⊕\Bbb Z_4⊕\Bbb Z_9$ Find an integer n such that $U(n)$ is isomorphic to $Z_2⊕Z_4⊕Z_9$
I have gotten this far:
I know $Z_2$ is isomorphic to $U(4)$ and $Z_4$ is isomorphic to $U(5)$. However, I'm having trouble figuring out what $Z_9$ is isomorphic to in regar... | $\mathbb{Z}_2\oplus \mathbb{Z}_4 \oplus \mathbb{Z}_9\cong\mathbb{Z}_2 \oplus \mathbb{Z}_{36}.$
$2+1$ and $36+1$ are distinct primes, so $\mathbb{Z}_3\oplus \mathbb{Z}_{37}\cong \mathbb{Z}_{3.37}$ would work.
Similarly $\mathbb{Z}_2\oplus \mathbb{Z}_4\oplus \mathbb{Z}_9\cong \mathbb{Z}_4\oplus \mathbb{Z}_{18}$.
Then $4+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1526588",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving df is the sum of partial derivatives Wikipedia states:
Since any vector $v$ is a linear combination $\sum v_je_j$ of its components, $df$ is uniquely determined by $df_p(e_j)$ for each $j$ and each $p \in U$, which are just the partial derivatives of $f$ on $U$. Thus $df$ provides a way of encoding the partial... | Let $A$ be open in $\Bbb{R}^{n}$; let $f: A \to \Bbb{R}$ be of class $C^{1}$; for every $x \in A$ let $df^{x}: v \mapsto \nabla f(x)\cdot v$ on $\Bbb{R}^{n}$. Then $df$, the differential of $f$, is a $1$-form on $A$ (provided that a $1$-form is defined as an alternating $1$-tensor).
Each elementary 1-form $dx_{i}$, whi... | {
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"timestamp": "2023-03-29T00:00:00",
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How many ways can you get a bag of $12$ donuts if a shop sells $30$ kinds of donuts? A donut shop sells $30$ kinds of donuts, and there are at least $12$ donuts of each kind. How many ways can you:
*
*Get a bag of $12$ donuts?
*Get a bag of $12$ donuts if you want at least $3$ glazed donuts and at least $4$ chocola... | Problem 1 is standard Stars and Bars. You solved it correctly. The number of ways is $\binom{30+12-1}{12}$, or equivalently $\binom{30+12-1}{30-1}$.
For Problem 2, I think we are to assume that glazed is one of the $30$ kinds, and that all glazed doughnuts (from that store) are identical. Ditto for chocolate.
So if we ... | {
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"url": "https://math.stackexchange.com/questions/1526808",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "4",
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Improper integrals - where am I going wrong? The question asks:
$\int_0^\pi \frac{sinx}{\sqrt{|cosx|}}$
My attempt:
$\lim\limits_{t \to \frac{\pi}{2}^-} \int_0^t \frac{sinx}{\sqrt{|cosx|}}$ + $\lim\limits_{t \to \frac{\pi}{2}^+} \int_t^\pi \frac{sinx}{\sqrt{|cosx|}}$
$\int \frac{sinx}{\sqrt{|cosx|}} = -2\sqrt{|cosx|}... | Note that after $\frac {\pi} 2$, cosine is negative, so the absolute value function in your second integral will have a -cos(x) in the bottom. That'll add an extra - to your antiderivative of the second part, which flips it from -2 to 2
| {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Does an inverse exist for a set of complex numbers with operation of multiplication? I will give some context of my question: I was trying to solve this problem
Let $S$ be a finite non-empty set of non-zero complex numbers which is closed under multiplication. Show that $S$ is a subset of the set $\{z \in \mathbb C: |... | To prove this in the order suggested:
*
*Let $a \in S$. If $|a|>1$, then $1 < |a| < |a|^2 < \cdots $ and the powers of $a$ would be an infinite subset of $S$. The same would follow if $|a|<1$. Thus, $|a|=1$.
*Let $S$ have $n$ elements. Consider $x \mapsto ax$. This is injective and so is surjective. Therefore $a_1 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1527082",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Calculating of $\lim_{x\to 0^+} x^{(x^x-1)}$ When I want calculate $\lim_{x\to 0^+} x^{(x^x-1)}$ , I have to calculate $\lim_{x\to 0^+}(x^x-1)\ln x$ . I calculat $\lim_{x\to 0^+}x^x=\lim_{x\to 0^+}e^{x\ln x}=1$ so $\lim_{x\to 0^+}(x^x-1)\ln x$ is $0\times \infty$ .
How can I calculate $\lim_{x\to 0^+}(x^x-1)\ln x$?
| $$\lim_{x\to 0^+} \exp((x^x-1)\ln x) =$$
$$\exp(\lim_{x\to 0^+} (x^x-1)\ln x) = \exp(\lim_{x\to 0^+} \frac{x^x-1}{\frac1{\ln x}}) = \exp(\lim_{x\to 0^+} \frac{\exp(x\ln x)-1}{\frac1{\ln x}})$$
Using Hopital
$$\exp(\lim_{x\to 0^+} \frac{\exp(x\ln x)-1}{\frac1{\ln x}})=\exp(\lim_{x\to 0^+} \frac{(\ln x+1)\exp(x\ln x)}{\f... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"answer_id": 1
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Integration by parts of Wirtinger derivative As a physics student, I am currently trying to use Wirtinger derivatives in everyday work.
While the basics are clear and so far seem to make life considerably simpler, I am having trouble to see if integration by parts is possible as in the real case.
Transforming to real t... | First of all, there is no way that your suggested formulas can be true. Your integrals over $D$ contain $1$-forms, which is nonsense since you're integrating over the 2-dimensional $D$.
On the other hand, Stokes' theorem is still valid in the complex setting:
$$
\int_{\partial D} \omega = \int_{D} d\omega.
$$
Furthermo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1527295",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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Conceptually, why does a positive definite Hessian at a specific point able to tell you if that point is a maximum or minimum? This is not about calculating anything. But can anyone tell me why this is the case?
So, from wikipedia:
If the Hessian is positive definite at x, then f attains a local minimum at x. If the H... | This is because of Taylor's formula at order $2$:
\begin{align*}f(x+h,y+k)-f(x,y)&=hf'_x(x,y)+kf'_y(x,y)\begin{aligned}[t]&+\frac12\Bigl(h^2f''_{x^2}(x,y)+2hkf''_{xy}(x,y)\\&+k^2f''_{y^2}(x,y)\Bigr)+o\bigl(\bigl\lVert(h,k)\bigr\rVert^2 \bigr)\end {aligned}\\
&=\frac12\Bigl(h^2f''_{x^2}(x,y)+2hkf''_{xy}(x,y)+k^2f''_{y^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1527379",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "14",
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"answer_id": 2
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Proof: 1007 can not be written as the sum of two primes. The claim is: 1007 can be written as the sum of two primes. We want to prove or disprove it.
Edit: My professor provided this definition in his previous assignment: An integer $n \geq 2$ is called prime if its only positive integer divisors are $1$ and $n$.
I ... | $1007$ is an odd number so it cannot be the sum of two odd numbers and it cannot be the sum of two even numbers. Therefore, it can only be the sum of an even and an odd number. Since $2$ is the only even prime it would have to be $2+1005$ and $1005$ is not prime.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1527778",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 4,
"answer_id": 3
} |
Largest irreducible representation of a finite non-commutative group Let $G$ be a finite non-commutative group of order $k$. Is there any way to determine a number $m$ such that there will necessarily exist an irreducible representation of $G$ of dimension $d \geq m$? If not, what kind of additional information could g... | As the comments indicate, the example of the dihedral groups shows that you cannot expect irreps of dimension greater than $2$ without additional hypotheses, and that bounding the size of the abelianization isn't a particularly helpful hypothesis.
One thing to say is that if a group $G$ has $c(G)$ conjugacy classes an... | {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "4",
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$S(x)=\frac{x^4}{2\cdot 4}+\frac{x^6}{2\cdot 4\cdot 6}+\frac{x^8}{2\cdot4\cdot6\cdot8}+\cdots$ Find the sum of $$S(x)=\frac{x^4}{2\cdot 4}+\frac{x^6}{2\cdot 4\cdot 6}+\frac{x^8}{2\cdot4\cdot6\cdot8}+\cdots$$
What I did so far:
It's trivial that
$$S'(x)-xS(x)=\frac{x^3}{2}-\lim_{n\to\infty}\frac{x^{2n}}{n!2^{{n(n+1)}/2}... | @labbhattacharjee provided a very efficient approach to evaluating the series.
The approach in the OP did not provide a way forward since the solution to the ODE $S'(x)-xS(x)=\frac12x^3$, $S(0)=0$ is given by
$$\begin{align}
S(x)&=\int_0^x \frac12x'^3 e^{(x^2-x'^2)/2}\,dx' \tag 1\\\\
&=e^{x^2/2}-\frac12x^2-1
\end{ali... | {
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"timestamp": "2023-03-29T00:00:00",
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Intermediate value theorem: showing there exists $c \in [0,2/3]$ such that $f(x+1/3) = f(x)$ Intermediate value theorem: show there exists $c \in [0,2/3]$ such that $f(c+\frac{1}{3}) = f(c)$
Let $f: [0,1] \to [0,1]$ be continuous and $f(0) = f(1)$
This is pretty straight forward and the other ones ive done it was easie... | HINT:
You were on the right track by defining $g(x)\equiv f(x+1/3)-f(x)$ for $x\in [0,2/3]$.
Now, note that $$g(0)+g(1/3)+g(2/3)=f(1)-f(0)=0$$Then either $g(0)=g(1/3)=g(2/3)=0$ or at least one of the three is positive and one of the three is negative. Now, use the IVT.
| {
"language": "en",
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Image of normal subgroup under surjective homomorphism as kernel Let $f: G \to H$ be a group homomorphism. Then the preimage $f^{-1}(N)$ of a normal subgroup $N$ of $H$ is normal in $G$: this preimage is the kernel of the composition of $f$ with the canonical projection $H \to H/N$.
Now, if $K$ is a normal subgroup in... | We want to push, by some homomorphism, $f(K)$ to identity (and no other elements to identity).
How can we do this: first pull-back $f(K)$ via (inverse of) $f$: sine $\ker(f)=N$, we have
$$f^{-1}(f(K))=KN \,\,\,\,(\mbox{which is normal in $G$})$$
Next, push this $KN$ (and only that) to identity via a homomorphism; the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1528367",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to prove this bound of $L^\infty$ norm. A differentiable function $ f:\mathbb R\to \mathbb R$ satisfies such conditions,
$ $\begin{cases} \lim_{x\to\infty} f(x)=\lim_{x\to-\infty} f(x)=0, &\\
\int_{-\infty}^{\infty}|{f(x)}|^{2}dx<\infty, \ \ \int_{-\infty}^{\infty}|{f'(x)}|^{2}dx<\infty. &
\end{cases}$$
Prove
$... | I'll assume $f'$ is continuous. Let $a\in \mathbb {R}.$ Then for $h>0,$ the FTC gives
$$f(a+h)^2/2-f(a)^2/2 = \int_a^{a+h} f\,f'$$
and
$$f(a-h)^2/2-f(a)^2/2 = -\int_{a-h}^{a} f\,f'. $$
Add these together, then take absolute values to get
$$|f(a+h)^2/2-f(a-h)^2/2 -f(a)^2| \le \int_{a-h}^{a+h} |f|\,|f'| \le \int_{\mathbb... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Gradient vector at some point Suppose you are given some function $f(x, y, z) = xy^3z^2$ and you are told to find its the gradient vector normal to the surface at some point (-1, -1, 2).
Is it right think that the gradient vector is normal to the surface as shown in the figure?
Also, if you substitute the point you ge... | Because you have a function $f(x,y,z)$ of three variables, I assume you're talking about the level surface at a certain point, where given a constant $k$ the function has the value
$$f(x,y,z)=k$$
Then in that case the gradient vector
$$\nabla f(x,y,z) =\frac{\partial f}{\partial x}\hat{\imath}+\frac{\partial f}{\partia... | {
"language": "en",
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Counting the number of words made of $2n$ letters
Compute the number of words made of $2n$ letters taken from the alphabet $\{a_1, a_2,\ldots,a_n\}$ such that each letters occurs exactly twice and no two consecutive letters are equal.
I started first of all thinking about the possibilities when we are not restricted ... | The strategy taken is to find the total number ways to arrange the letters $-$ the total number of ways to arrange the letter such that at least one pair of letters are together.
The total number of ways to arrange the letter as you have found is: $$\frac{(2n)!}{2^n}$$
Let $A_i$ be the number of ways to arrange the $2n... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How are theses functions called and what are their properties? We define a system of two equations with two variables and six parameters:
$\begin{cases}
a_1x+b_1y=c_1\\
a_2x+b_2y=c_2
\end{cases}$
We obtain a function $f:\mathbb{R^3\times\mathbb{R}^2\rightarrow\mathbb{R}^2}$,
$$\pmatrix{a_1 & a_2 \\ b_1 & b_2 \\ c_1 & ... | The functions in OP are defined by the system:
$$\begin{cases}
a_1x+b_1y=c_1\\
a_2x+b_2y=c_2
\end{cases}$$
so that:
$$
x=\dfrac {\Delta_x}{\Delta} \qquad y=\dfrac {\Delta_y}{\Delta}
$$
where:
$$
\Delta=a_1b_2-a_2b_1 \quad \Delta_x=b_1c_2-b_2c_1 \quad \Delta_y=a_1c_2-a_2c_1
$$
so that the functions $f_{c_1}:\mathbb{R}\t... | {
"language": "en",
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Structure of dense subset of $\mathbb{R}$ Here user Mario Carneiro formulated statement which I find really interesting and useful and I proved it and I would like to know is my proof correct?
If sequence $\{a_n\}_{n=1}^{\infty}$ implies the following conditions:
1) $a_1<a_2<\dots<a_{n}<\dots$
2) $a_n\to \infty$ as... | There are just two very small corrections that need to be made. First, there is a $k\in\Bbb N$ such that $a_k\color{red}{\le}x<a_{k+1}$. And secondly, in $(2)$ there is an $M\in\Bbb N$ such that $x+M\in\color{red}[a_\ell,a_{\ell+1})$ for some $\ell\ge N$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Two non-differentiable functions whose product is differentiable. So I was wondering while studying analysis if there is any case where two functions aren't differential at $0$ (kind of like $1/x$) but is differentiable at 0 when combined (i.e. $fg$).
I mean this for functions that are defined on $\mathbb{R}$.
| On $\mathbb R$ let $f$ be the indicator of $[0,\infty)$ and $g$ the indicator of $(-\infty,0)$. Their product is $0$, whereas each of the functions is not differentiable at $0$.
Furthermore, you dont need your functions to be defined on $\mathbb R$ to say something about differentiability in $0$, since differentiabilit... | {
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Linear Transformations, Linear Algebra Let T:P3→P3 be the linear transformation such that $T(−2x^2)= −2x^2 − 2x$, $T(0.5x + 2)= 3x^2 + 4x−2$, and $T(2x^2 − 1)= 2x + 1$.
Find $T(1), T(x), T(x^2)$, and $T(ax^2 + bx + c)$, where a, b, and c are arbitrary real numbers.
I understand how to find $T(x^2)$ where you just divi... | We can write $1=-(-2x^2)-(2x^2-1), x=2(0.5x+2)-4$, so
$$T(1)=T(-(-2x^2)-(2x^2-1))=-T(-2x^2)-T(2x^2-1)=2x^2-1$$
$$T(x)=T(2(0.5x+2)-4)=2T(0.5x+2)-4T(1)=-2x^2+8x$$
Then we get
$$T(ax^2+bx+c)=a(x^2+x)+b(-2x^2+8x)+c(2x^2-1)$$
$$=(a-2b+2c)x^2+(a+8b)x-c$$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Remainder of the polynomial division, knowing other remainders Let $p(x)$ be a polynomial of 3rd degree.
We know that the division of $p(x)$ by $x-4$ gives us a remainder of 2 and divided by $x+2$ gives us the remainder of 1.
What's the remainder of $p(x)$ by $(x-4)(x+1)$?
I've used the remainder theorem but I don't se... | We can write
$$p(x)=(x-4)(ax^2+bx+c)+2.$$
Since $p(-2)=1$, we have
$$1=(-6)(4a-2b+c)+2\quad\Rightarrow\quad c=-4a+2b+\frac 16.$$
So, we can write
$$p(x)=ax^3+(b-4a)x^2+\left(-4a-2b+\frac 16\right)x+16a-8b+\frac 43.$$
Thus, we have
$$p(x)=(x-4)(x+1)(ax+b-a)+\color{red}{\left(-3a+b+\frac 16\right)x+12a-4b+\frac 43}.$$
T... | {
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"timestamp": "2023-03-29T00:00:00",
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Does the order of operations matter with just addition and subtraction? Had a debate on whether you could do addition/subtraction in any order you want. Specifically, for the following:
$9 - 4 + 3$
We both agree that the answer is 8.
I argue that, by giving addition a higher priority than subtraction (rather than the s... | I would say that the order (for adding/subtracting) doesn't matter AND you need not elevate 'priorities' so long as you realize that subtracting is just adding negative numbers...
So the 9 - 4 + 3 isn't 9 - (4 + 3) but rather is 9 + ( -4 + 3 ).
Putting the negative sign in front of the parenthesis implies that you wan... | {
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$G$ is a group, if ∀a,b∈G, $a^2b=ba^2$, is $G$ Abelian? $G$ is a group, if ∀a,b∈G, $a^2b=ba^2$, how to make a counterexample show that $G$ is NOT Abelian?
What's the counterexample when $a^nb=ba^n$?
| To get a counterexample, you need an element which is not the square of another element. A good example would be the quaternion group, where $i^2=j^2=k^2=-1$, and $(-1)^2=1$. $-1$ commutes with all the elements in the quaternion group.
Read the Wikipedia article here.
Since every element in the quaternion group raised ... | {
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"timestamp": "2023-03-29T00:00:00",
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Find the minimal polynomial for the given matrix. The first part is trivial. How to go about the second part?
| You can check that
$$A^2=\begin{pmatrix}
0 &c &ac\\
0&b&ab+c\\
1&a&a^2+b
\end{pmatrix}
$$
Suppose that the minimal polynomial is $m(x) = x^2+\alpha x+\beta$. Then $m(A) = 0$. Notice that
$$(m(A))_{11} = 0+0+\beta=0\Rightarrow \beta=0$$
$$(m(A))_{21} = 0+\alpha+0=0\Rightarrow \alpha=0$$
Thus, $A^2=0$ which is clearly f... | {
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Calculating syzygies with Macaulay2 I'm trying to calculate the syzygies of a set of elements on the polinomial ring of 6 variables. But I'm trying to specify the number of generator in each degree the syzygies have. I know that Macaulay2 can give me the syzygies of the sistem very fast, but it is returning some column... | I think the problem here is understanding Macaulay2's notation. Let's take an example:
i1 : R = QQ[x,y,z]
o1 = R
o1 : PolynomialRing
i7 : I = ideal(random(2,R), random(3,R), random(2,R));
o7 : Ideal of R
i8 : betti res I
0 1 2 3
o8 = total: 1 3 3 1
0: 1 . . .
1: . 2 . .
2: .... | {
"language": "en",
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"source": "stackexchange",
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Area between three curves involves tangent line Let $R$ be the region enclosed by the $x$-axis, the curve $y=x^{2}$, and the tangent to the curve at $x=a$, where $a>0$. If the area of $R$ is $\frac{2}{3}$, then the value of $a$ is?
Is there any clue for the tangent equation and how's the integral equation will be?
Than... | Hints :
tangent is passing through $(a,a^2)$ where its slope would be $\displaystyle{\frac{dy}{dx}|_{x=a}}$
Area of R would be the difference between the region of curve and the region of tangent line enclosed by the x-axis. (from $0$ to $a$)
| {
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Covering a rectangle with circles On a rectangle table with area A, n unit-radius circles are placed and it is not possible to place any extra circles without overlapping with some of the existing ones or without placing circle's center outside the rectangle which results in falling off of the table.
Notice that this ... | WOLOG, choose the coordinate system so that the table cover the rectangle
$$R = [0,w] \times [0,h]\quad\text{ with }A = wh.$$
Let $C = \{ \vec{c}_1, \vec{c}_2, \ldots, \vec{c}_n \}$ be the centers of the $n$ circles.
For any $\vec{x} \in R$, let
$$d_C(\vec{x}) = \min\{ d(\vec{x},\vec{c}) : \vec{c} \in C\}$$
be the mi... | {
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permutation/combination There are six cards with numbers written on them: $1, 2, 3, 4, 5, 6$ .
Two of them are drawn at at time and put together to form fractions, e.g. $\dfrac45$
How many proper fractions can be formed?
What is correct equation to solve this type of questions?
thanks
| If you think of all the possible fractions, exactly half of them will be proper (since you're always drawing cards of two different values), so there are $6 \times 5 \times \frac12$ possibilities.
| {
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Symmetric group isomorphic to semidirect product of Alternating group and Z/2Z
I'm having a hard time understanding why $A_n \rtimes \mathbb{Z}_2 \cong S_n$.
I understand that $A_n$ is normal in $S_n$. But that's about it. What would the $\alpha$: $\mathbb{Z}$$_2$$\longrightarrow$Aut($A_n$) be?
| Pick an odd element of $ S_n $ of order two; for example, a transposition. Then conjugation by that element is an order two automorphism of $ A_n $.
| {
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Is every manifold a metric space? I'm trying to learn some topology as a hobby, and my understanding is that all manifolds are examples of topological spaces. Similarly, all metric spaces are also examples of topological spaces. I want to explore the relationship between metric spaces and manifolds, could it be that al... | Assuming the usual definition of a topological manifold as a locally Euclidean space which is both Hausdorff and second-countable, it turns out that every manifold $M$ is a metrizable space. That is, you can put a metric on $M$ which induces the topology of $M$. This follows from example from Urysohn's metrization theo... | {
"language": "en",
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Finding the value of this limit $$\lim_{x\to 0}\frac{\sin x^4-x^4\cos x^4+x^{20}}{x^4(e^{2x^4}-1-2x^4)}$$
The answer is given to be $1$. Can someone give any hint/s related to this problem?
More importantly, why am I not able to get the answer by simply using L hopital's rule and by basic substitution of trigonometric... | Divide numerator and denominator by $x^4$, giving you
$$\lim_{x\to 0}\frac{\frac{\sin x^4}{x^4}-\cos x^4+x^{16}}{e^{2x^4}-1-2x^4}$$
Now expand the sine, cosine, and exponential into power series (Maclaurin series) with as many terms as needed.
| {
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Find the minimum of $(u-v)^2+ (\sqrt{2-u^2}-\frac{9}{v})^2$ for $00$.
Find the minimum of $\displaystyle (u-v)^2+ \left(\sqrt{2-u^2}-\frac{9}{v}\right)^2$ for
$0<u<\sqrt{2}$ and $v>0$.
I think I have to use the Arithmetic and Geometric Means Inequalities. Or $\displaystyle \frac{1}{2}(u-v)^2+ \left(\sqrt{2-u^2}-\fr... | You can simply think $(u-v)^2+ (\sqrt{2-u^2}-\frac{9}{v})^2$ is the square of distance between two points $a$ and $b$ in the plane, where
$$a=(u,\sqrt{2-u^2}),b=(v,\frac{9}{v}).$$
Clearly, $a$ satisfies $x^2+y^2=2,(0<x<\sqrt{2})$ and $b$ satisfies $y=\frac{9}{x},(x>0)$. Then from the geometry graph, we easily know the ... | {
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How to prove D70 = {1, 2, 5, 7, 10, 14, 35, 70} is a Boolean algebra Prove that the set $D_{70}$ = {1, 2, 5, 7, 10, 14, 35, 70} of positive factors is a Boolean algebra under the operation (+), (.), (') defined by
$$x + y = lcm(x, y)$$
$$x . y = gcd(x, y)$$
$$x' = \frac{70}{x}$$
Attempt:
To Prove $D_{70}$ is a boolean... | Write out the operation tables for "+", ".", and " ' ".
Then check that for all x, y, z in D70 under those operations that the distributive laws hold.
You could write a computer program for the last part, use OTTER, or treat those tables as an 8-valued logical system and write out truth tables for the computation.
| {
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Arithmetic sequence to geometric sequence.
The numbers $a_1, a_2, a_3, . . .$ form an arithmetic sequence with $a_1 \ne a_2$. The three
numbers $a_1, a_2, a_6$ form a geometric sequence in that order. Determine all possible
positive integers $k$ for which the three numbers $a_1, a_4, a_k$ also form a geometric
s... | HINT :
Use the fact that
$$\text{$a,b,c$ form a geometric sequence}\Rightarrow b^2=ac$$
Thus, you have
$$a_2^2=a_1a_6\quad\text{and}\quad a_4^2=a_1a_k$$
Express these by $a_1$ and $d$.
| {
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How can I change the variable in this integral? I have the following equality
$\int_{0}^{\pi}cos(nt)cos(mt)dt=0$ (if $m\neq n$)
This is in the context of Chebyshev polynomials and the book states the following to deduce ortoghonality.
"the orthogonality property is drawn by using the substitution t = arcos(x) in the i... | $t = \arccos{x} \implies dt = -(1-x^2)^{-1/2} \, dx$.
Also note that
$$T_m(x) = \cos{(m \arccos{x})} $$
| {
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About the rank of submodules over PID
If $M$ is a free $R$-module of finite rank $n$ and $R$ is a PID, do
proper submodules of $M$ have strictly less rank than $M$?
I know that in this case, every submodule of $M$ is free and has finite rank $m\leq n$, but can we guarantee that if the submodule is proper, its rank ... | The rank of a submodule $N$ of $M$ will be equal to the rank of $M$ if and only if the quotient module $M/N$ is a torsion module, by the fundamental theorem of finitely generated modules over P. I. D.s. Hence its rank will be strictly less than the rank of $M$ if and only if $M/N$ contains non-torsion elements.
| {
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Product of numbers $\pm \sqrt{1} \pm \sqrt{2} \pm \dots \pm \sqrt{100}$ is a perfect square Let $A$ be the product of $2^{100}$ numbers of the form
$$\pm \sqrt{1} \pm \sqrt{2} \pm \dots \pm \sqrt{100}$$
Show that $A$ is an integer, and moreover, a perfect square.
I found a similar problem here, but the induction doesn... | The same argument can be used to show that the product of the $2^{n-1}$ numbers $\sqrt{1} \pm \sqrt{2} \pm \cdots \pm \sqrt{n}$ is an integer.
Then your number is exactly the square of this product.
| {
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"timestamp": "2023-03-29T00:00:00",
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Obtain the expression $T=\frac x2+\sum_{n=2}^\infty (-1)^{n-1} \frac{1*3*\ldots*(2n-3)}{n!}*\frac{x^n}{2^n}$ Obtain the expression
$$T=\frac x2+\sum_{n=2}^\infty (-1)^{n-1} \frac{1*3*\ldots*(2n-3)}{n!}*\frac{x^n}{2^n}$$
for one root of the equation
$$T^2+2T-x=0,$$
and show it converges so long as $|x| \lt 1$.
I have ... | First notice that
\begin{align}
T(x) &= \frac{x}{2} - \frac{1}{2} \, \sum_{n=2}^{\infty} (-1)^{n-1} \frac{(-1/2)_{n} \, x^{n}}{n!} \\
&= \frac{x}{2} + \frac{1}{2} \, \left[ (1+x)^{-1/2} - x -2 \right] \\
&= -1 + \sqrt{1+x}
\end{align}
Now the quadratic equation $T^{2} + 2 \, T - x = 0$ leads to
$$T(x) = - 1 \pm \sqrt... | {
"language": "en",
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Irreducible finite-state Markov Chain Could anyone help me with this?
Let $X_n$ be an irreducible Markov chain on the state space $S=\{1,\ldots,N\}$. Show that there exist $C<\infty$ and $\rho <1$ such that for any states $i,j$,
$ P (X_m \neq j, m = 0, \ldots, n | X_0 =i) \leq C \rho^n.$
Show that this implies that $E... | Use this fact that if process has not hitted $j$ in $m$ steps, it's not hitted $j$ in $k$ steps where $k$ is greatest multiple of $n$ smaller than $m$.
In fact, if there is a $\delta$ such that beginning from each $i$, you have reached $j$ in $n$ (may be smaller) steps with probablity greater than $\delta$, then take $... | {
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Why does count of Z -Transform of sequence change? I was looking at the reference [1] below and noted the author defined the Z-transform for [1, 2, 3] as $$[6, \frac{11}{4}, 2]$$
I worked it out as follows:
$$X[z]=\sum_{n=0}^2x[n]z^{-n}$$
$$=x[0]z^0+\frac{x[1]}{z}+\frac{x[2]}{z^2}$$
$$=1+\frac{1}{z}+\frac{2}{z^2}$$
so ... | When I wrote that post I used 0-based indexing for the input because that's just what programmers do by default, and I used 1-based indexing for the output because of the division-by-zero issue. I don't know if it's more common for the indices to start at 1 for the input.
Basically, don't worry too much about it. It's ... | {
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How to solve $\max|x_1|\Sigma_{i=1}^n|x_i|$, with constraint $\Sigma_{i=1}^n|x_i|^2=1,n>2$ Alternatively, that making the upper bound for $|x_1|\Sigma_{i=1}^n|x_i|$ as tight as possible is also welcome. For me, if $\Sigma_{i=1}^n|x_i|=\sqrt{n}$, then $|x_1|=\sqrt{1/n}$. If $|x_1|=1$,then $\Sigma_{i=1}^n|x_i|=1$. In a w... | WLOG, $x_i\geq 0$ for every $i$. From $\sum_{i>1}x_i\leq \sqrt{(n-1)\sum_{i>1}x_i^2}=\sqrt{(n-1)\left(1-x_1^2\right)}$, we have $x_1\sum_{i=1}^nx_i\leq x_1^2+x_1\sqrt{(n-1)\left(1-x_1^2\right)}$. By AM-GM, $\sqrt{ax_1^2\cdot b\left(1-x_1^2\right)}\leq \frac{b}{2}-x_1^2$, where $0\leq a<b$ satisfy $ab=n-1$ and $b-a=2$... | {
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What are the residues of $\frac{z^2 e^z}{1+e^{2z}}$? I hope to know the singularity and pole of $\frac{z^2 e^z}{1+e^{2z}}$.
I try $\frac{z^2 e^z}{1+e^{2z}} = \frac{z^2}{e^{-z}+e^{z}}$ and observe that the denominator seems like cosine function. So I think the singularites are i(2n+1)$\pi$/2. But when I try to evaluate ... | Since $\frac{z^2 e^z}{1+e^{2z}} = \frac{z^2}{2\cosh z}$, let's start with
$$f(z)=\frac{z^2}{2\cos iz}$$
Now both $z^2$ and $\cos iz$ are entire. The function can be singular only where $z_{0_n} = iz = \frac{\pi +2\pi n}{2}$ and since the zeroes of the denominators are first order, we have
$$\begin{align*}
Res f(z_{0_n}... | {
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What is the positive divisors of $n(n^2-1)(n^2+3)(n^2+5)$ I want to find the positive divisors of $n(n^2-1)(n^2+3)(n^2+5)$
from $n(n-1)(n+1)$ 2 and 3 should divide this expression for all positive n. how can I find the rest? which python says $(2, 3, 6, 7, 9, 42, 14, 18, 21, 63,126)$
| If $$n\equiv 0 \mod 3,$$ then the factor $$n^2+3\equiv 0 \mod 3.$$ Otherwise
$$n^2\equiv1 \mod 3$$ and thus $$n^2+5\equiv 0\mod3.$$
Therefore $$(n^2+3)(n^2+5)\equiv0\mod3.$$
Let us now look modulo $7$ and compute the number for all elements of $\mathbb Z_7$ (which means to check if the number is a multiple of $7$ for $... | {
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Prove or Disprove: if $\lim\limits_{n \to \infty} (a_{2n}-a_n)=0,$ then $\lim\limits_{n \to \infty} a_n=0.$ Prove or Disprove: if $\lim\limits_{n \to \infty} (a_{2n}-a_n)=0,$ then $\lim\limits_{n \to \infty} a_n=0.$
I don't think that this is true. and I'm trying to think about counterexample, but couldn't figure out a... | Just let $a_n = 1$ for all $n$, for example.
| {
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Is there another kind of two-dimensional geometry? I learned that in two-dimensional geometry, there are Euclidean geometry, hyperbolic geometry and spherical geometry. These geometries are homogeneous and isotropic. Is there another kind of two-dimensional geometry that is homogeneous and isotropic?
| In addition to the nine following a specific set of rules,
There are also more exotic versions such as projective geometry (think twisting the plane into a Mobius strip) and finite geometries.
If you relax the isotropic requirement you open up more possibilites such as the Manhattan metric aka Taxicab Geometry.
| {
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"url": "https://math.stackexchange.com/questions/1531772",
"timestamp": "2023-03-29T00:00:00",
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How to prove that $\frac1{n\cdot 2^n}\sum\limits_{k=0}^{n}k^m\binom{n}{k}\to\frac{1}{2^m}$ when $n\to\infty$ I have solve following sum
$$\sum_{k=0}^{n}k\binom{n}{k}=n2^{n-1}\Longrightarrow \dfrac{\displaystyle\sum_{k=0}^{n}k\binom{n}{k}}{n\cdot 2^n}\to\dfrac{1}{2},n\to\infty$$
$$\sum_{k=0}^{n}k^2\binom{n}{k}=n(n+1)2^{... | Let $(Z_i)$ be i.i.d. Bernoulli(1/2) random variables, and set $\bar Z_n={1\over n}\sum_{i=1}^n Z_i$ be the sample average. Then $\bar Z_n\to 1/2$ almost surely by the law of large numbers.
On the other hand, $X=\sum_{i=1}^n Z_i$ has a Binomial$(n,1/2)$ distribution, so
that, $\mathbb{P}(X=k)={n\choose k}(1/2)^n$ for ... | {
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Find the tight bound for the recurrence relation $T(n) = T(\frac{n}{3}) + 6^n $ $$T(1) = 2$$
$$T(n) = T(\frac{n}{3}) + 6^n \text{ For n > 1}$$
I tried using the substitution method to find it's closed form but I even from there, I could not figure out how to find its bound. My working:
$k = 1$
$$T(n) ... | Hint:
$$6^n > n^{\log_3 2}$$
Can you take another look at the Master theorem cases ?
| {
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} |
Common methods for proving: Existence proof For every real number x with $x\neq -1$ there exists a real number y such that $ \frac{y}{y+1}=x $.
$ \forall x\in \Bbb R\setminus (1) \ \ \exists y \in \Bbb R : \frac{y}{y+1}=x \\, x \neq -1 $
$ So \ \ if \ \ I \ \ choose \ \ e.g. \ x= \frac{1}{2} \ \ then \ \ y=1 \ \ such... | Use the transformation of $y\rightarrow z-1$ to change it to for all $x\neq 1$ there exists $z$ such that $\frac{z-1}{z}=1-\frac{1}{z}=x$, which is easily solvable for $z$ in terms of $x$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1532064",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Why the ordering of the quantifiers matters here? $\newcommand{\fool}{\operatorname{fool}}$I want to transfer the statement
You can fool all the people some of the time
into predicate logic.
Now I am using fool(X,T) to mean that I fool person X at time T.
Now I know that the correct form is $\exists T\ \forall X\ ... | To expand on AJ stas's comment, the first formulation, $\exists T \forall X fool(X,T)$, first you assert the existence of (at least one) single Time, T. Then you say that for EVERY person X, $fool(X,T)$, i.e, every single person can be fooled at that exact time.
When you reverse the quantifiers, $\forall X \ex... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1532293",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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What is $\langle a \rangle + \langle b \rangle$ where $a$ and $b$ are natural numbers? Let $\langle a \rangle$ and $\langle b \rangle$ be ideals in $\mathbb{Z}$, where a and b are natural numbers.
Define
$$S = \langle a \rangle + \langle b \rangle = \{x + y \mid x \in \langle a \rangle \text{ and } y \in \langle ... | For the other direction, let $x+y\in S$ where $s\in \langle a\rangle, y\in \langle b \rangle$. Then there are $h,k\in\mathbb{Z}$ such that $x = ha$, $y = kb$.
Then with $d = \gcd(a,b)$, $d$ is a divisor of $a$ and $b$, so there are $n$ and $m$ such that $a = nd$ and $b = md$. So $x + y = ha + kb = hnd + kmd = (hn + km)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1532416",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
once, twice, thrice differentiable So I am going through abbotts 2nd edition of Understanding analysis and I would like to know if this is correct.
Exercise:
(a) Let $g : [0, a]→R$ be differentiable, $g(0) = 0$, and $|g'(x)| ≤M$ for all $x∈[0, a]$. Show $|g(x)| ≤Mx$ for all $x∈[0, a]$.
(b) Let $h : [0, a]→R$ be twice d... | I only read your proof for (a), and it is slightly confusing. I think it might have the general idea right, but it is very unclear because of the poor presentation. You need to fix these issues:
*
*Currently, you say that by MVT on $[0,a]$, you know there exists some $c\in(0,x)$ such that $g(x)-g(0) = (x-0)\cdot g'(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1532542",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
finding density of $1/Z$ when $Z$ is a standard random normal variable If $Z$ is a standard normal r.v., we know that its density is
$$f_Z(z)=\frac{1}{\sqrt{2\pi}}e^{-z^2/2},$$
where $-\infty \leq z\leq \infty$. I want to find what $f_{1/Z}(z)$ is.
I let $Y=1/Z$, so $Z=1/Y$, and then I used the above formula to obtain ... | By the properties of a continuous density,
$$
\mathbb P(0 < Y \le b) = \int_0^b f_Y(y) \, dy.
$$
Therefore, for $Y = 1/Z$,
$$
\mathbb P(0 < Y \le y) = \mathbb P(Z \ge 1/y) = \int_{1/y}^{\infty} f_Z(z) \, dz = 1/2 - \int_0^{1/y} f_Z(z) \, dz.
$$
Differentiating with respect to $y$ to recover $Y$'s density,
$$
f_Y(y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1532650",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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How does this definition of curl make sense? In our maths textbook, curl is defined as $\text{curl} \textbf{F} = \nabla \times \textbf{F}$. If the $\nabla$ operator isn't a vector, but an operation on a vector, how can it be an operand of a cross product?
$$\textbf{F} \in \mathbb R^3 \\ (\nabla) \in \mathbb R^3 \right... | $
\mathrm{\nabla}
$
can be regard as a vector
$
\left({\frac{}{\mathrm{\partial}{x}}\mathrm{,}\frac{}{\mathrm{\partial}{y}}\mathrm{,}\frac{}{\mathrm{\partial}{z}}}\right)
$
to participate in calculation
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1532765",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Is every Banach space densely embedded in a Hilbert space? Can every Banach space be densely embedded in a Hilbert space? This is clear if the Banach space is actually a Hilbert space, but much can you relax this?
If the embedding exists, is the target Hilbert space unique?
| I will show that the answer is yes, for any separable Banach space $B$. The non-separable case is covered by George Lowther above.
So let $B$ be a separable Banach space, and consider some countable dense subset $\{x_n\}_{n\in \Bbb N} \subset B$. For each $n$ we apply the Hahn Banach theorem to find some $f_n \in B^*$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1532856",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "25",
"answer_count": 4,
"answer_id": 0
} |
Proving the determinant of a matrix is positive How can I show that $\det A \leq 0 $,
if $A = \begin{pmatrix}
a & b & c\\
b & d & e\\
c & e & f
\end{pmatrix}$
and $a>0, ad-b^2 = 0$
| We have $$|A|=adf-b^2f-ae^2-dc^2+2bce=-(ae^2+dc^2-2bce)=-(ae^2+dc^2\pm 2\sqrt{ad}ce)=-(e\sqrt{a}\pm c\sqrt{d})^2\le 0$$
Note that we need $a>0$ for the step $\sqrt{ad}=\sqrt{a}\sqrt{d}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1532952",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Arranging 6 different people in all possible ways in a 3 floor house Say we have a house that fits 6 people. All the rooms on the left side of the house have big rooms, all the rooms on the right side have small rooms
---------
Floor 3| A | B |
---------
Floor 2| A | B |
---------
Floor 1| A | B |
... |
So it seems that since every floor's permutation counts as only one ordering, the answer for this house arrangement is $\frac{6!}{2!}$
This part of the answer is wrong. Since there are 3 floors, on each of which the permutations don't matter, the denominator will change as:
$$\frac{6!}{2!^3}$$
Similarly:
For this ho... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1533045",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
$\mathbf{E}[X|Z=z]=\mathbf{E}[Y|Z=z]$ implies $X=Y$ a.s.? Suppose that $X,Y,Z$ are real-valued random variables defined on a probability space $(\Omega, \mathscr{F}, P)$ such that for all reals $z$ such that $P(Z=z)>0$ it holds
$$
\mathbf{E}[X|Z=z]=\mathbf{E}[Y|Z=z].
$$
What can be said about $X$ and $Y$? In particular... | Just consider $X, Z$ and $Y, Z$, to be independent, $X$ and $Y$ can have any probability distribution as long as they have equal mean. Then your equality holds for all $z$, but $X$ and $Y$ are not a.s. equal.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1533141",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Determine which of the following relation is a function? Given two set $ A = \{0, 2, 4, 6\}$ and $B = \{1, 3, 5, 7\}$, determine which of the following relation is a function?
$(a) \{(6, 3), (2, 1), (0, 3), (4, 5)\}$,
$(b) \{(2, 3), (4, 7), (0, 1), (6, 5)\}$,
$(c) \{(2, 1), (4, 5), (6, 3)\}$,
$(d) \{(6, 1), (0, 3), (4,... | A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.
So, according definition of function, given relations with domain $A = \{0, 2, 4, 6\}$,
$(a) \{(6, 3), (2, 1), (0, 3), (4, 5)\} = \{ (0, 3),(2, 1), (4, 5),(6, 3)\}$ is fu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1533226",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Homework: Sum of the cubed roots of polynomial Given $7X^4-14X^3-7X+2 = f\in R[X]$, find the sum of the cubed roots.
Let $x_1, x_2, x_3, x_4\in R$ be the roots. Then the polynomial $X^4-2X^3-X+ 2/7$ would have the same roots. If we write the polynomial as $X^4 + a_1X^3 + a_2X^2 +a_3X + a_4$ then per Viete's theorem:
$a... | Hint: as we can have only cubic terms in the symmetric polynomial sums, the only terms which can be used are of form $(\sum x)^3, \sum x \sum xy$ and $\sum xyz$. Then it is a matter of testing $3$ coefficients...
$$\sum_{cyc} x_1^3 = \left(\sum_{cyc} x_1 \right)^3-3\left(\sum_{cyc} x_1 \right)\left(\sum_{cyc} x_1 x_2 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1533363",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Laplace transform to bio heat equation This is the bio heat equation and I have several questions when trying to work with it.
$$
\rho c \frac{\partial u(x,t)}{\partial t}
= \nabla[k \nabla u(x,t)]
+ \omega_b \rho_b c_b [u_a - u(x,t)]
+ Q_m + Q_r(x,t)
$$
where the first expression on the right hand side describes... | *
*I think yes if $$\nabla[k \nabla u(\underline x,t)] = \nabla \cdot [k \nabla u(\underline x,t)]:$$
By one of the product rules (),
$$\nabla[k \nabla u(\underline x,t)]$$
$$= \nabla k \nabla u(\underline x,t) + k \nabla \nabla u(\underline x,t)]$$
$$= (0) \nabla u(\underline x,t) + k \nabla \nabla u(\underline x,t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1533458",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to show that $a_{1}=a_{3}=a_{5}=\cdots=a_{199}=0$ Let $$(1-x+x^2-x^3+\cdots-x^{99}+x^{100})(1+x+x^2+\cdots+x^{100})=a_{0}+a_{1}x+\cdots+a_{200}x^{200}$$
show that
$$a_{1}=a_{3}=a_{5}=\cdots=a_{199}=0$$
I have one methods to solve this problem:
Let$$g(x)=(1-x+x^2-x^3+\cdots-x^{99}+x^{100})(1+x+x^2+\cdots+x^{100})$$
... | Let $A (x) = (1-x+x^2-x^3+\cdots-x^{99}+x^{100})$ and $B (x) = 1+x+x^2+\cdots+x^{100}$. Then we have $g(x) = A (x) B (x)$. We have $(x + 1) A (x) = x^{101} + 1$ and $(x - 1) B (x) = x ^ {101} - 1$. Then $$g (x) = \frac {x^ {202} - 1} {x^2 - 1} = x^{200} + x^{198} + \cdots + 1,$$ as desired.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1533555",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 2
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Complicated sum with binomial coefficients I know how to prove, that $\frac{1}{2^{n}}\cdot\sum\limits_{k=0}^nC_n^k \cdot
\sqrt{1+2^{2n}v^{2k}(1-v)^{2(n-k)}}$ tends to 2 if n tends to infinity for $v\in (0,\, 1),\ v\neq 1/2$. This can be proved with the use of Dynamical systems reasonings, which are non-trivial and comp... | You can rewrite the expression as
$$
\sum_{k=0}^n \sqrt{P_{1/2,n}(k)^2 + P_{v,n}(k)^2},
$$
where $P_{a,n} = {n\choose k} a^k (1-a)^k = P\big(\mathrm{Binom}(n,a) = k\big)$ is the binomial distribution with parameters $n$ and $a\in(0,1)$.
Now the statement is pretty clear, since by the law of large numbers the distribu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1533640",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 2,
"answer_id": 1
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Solve $x-\lfloor x\rfloor= \frac{2}{\frac{1}{x} + \frac{1}{\lfloor x\rfloor}}$ Could anyone advise me how to solve the following problem:
Find all $x \in \mathbb{R}$ such that $x-\lfloor x\rfloor= \dfrac{2}{\dfrac{1}{x} + \dfrac{1}{\lfloor x\rfloor}},$ where $\lfloor *\rfloor$ denotes the greatest integer function.
He... | Completing the square gives
$$
x^2-2x\lfloor x\rfloor+\lfloor x\rfloor^2=2\lfloor x\rfloor^2
$$
so
$$
(x-\lfloor x\rfloor)^2=2\lfloor x\rfloor^2
$$
Since $0\le x-\lfloor x\rfloor<1$, we conclude $\lfloor x\rfloor=0$ that's disallowed by the starting equation.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1533765",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
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Convergence of a series with binomial coefficient Let $S_n$ be a series with binomial coefficients as follows:
$$ S_n = \sum_{k=1}^n \begin{pmatrix} n \\ k \end{pmatrix}\frac 1k\left(-\frac{1}{1-a} \right)^k\left(\frac{a}{1-a} \right)^{n-k},$$
where $0 < a < 1$. My question is: when $n\to +\infty$, does the series $S_n... | We have:
$$\begin{eqnarray*} S_n = \frac{1}{(1-a)^n}\sum_{k=1}^{n}\binom{n}{k}\frac{(-1)^k}{k}a^{n-k}&=&\frac{1}{(1-a)^n}\sum_{k=1}^{n}\binom{n}{k}a^{n-k}\int_{0}^{-1}x^{k-1}\,dx\\&=&\frac{1}{(1-a)^n}\int_{0}^{-1}\frac{dx}{x}\sum_{k=1}^{n}\binom{n}{k}x^k a^{n-k}\,dx\\&=&\frac{1}{(1-a)^n}\int_{0}^{-1}\frac{(a+x)^n-a^n}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1533885",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Determining whether a transformation is linear I was hoping someone would be able to explain the following to me as I do not understand it:
f is the reflection in the line x = 1.
The reflection f maps the origin to point (2, 0) so it does not fix the origin. Therefore f is not a linear transformation.
Would someone b... | Here is a visualization:
How does the reflection transformation map the origin?
$f(P) = P'$ such that the line segment $PP'$ is halved by the mirror line.
The shortest distance to the mirror line $x = 1$ is the same for point $P$ and mirrored point $P'$.
Why is this transformation not linear?
Linear transformations h... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1534081",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Probability of symmetric difference inequality I think I am missing something easy here, but my book notes that
$$P\left\{\bigcup_{j=1}^\infty A_j \mathbin\triangle \bigcup_{j=1}^\infty B_j\right\} \leq \sum_{j=1}^\infty P\{A_j \mathbin\triangle B_j\}$$
where $A\mathbin\triangle B$ is the symmetric difference of the tw... | Well, let's look at the base case.
$$\begin{align}(A_1\cup A_2)\triangle(B_1\cup B_2)
& = ((A_1\cup A_2)\cap B_1^c \cap B_2^c)\cup(A_1^c\cap A_2^c\cap(B_1\cup B_2))
\\[1ex] & = (A_1\cap B_1^c\cap B_2^c)\cup(A_1^c\cap A_2^c\cap B_2)\cup (A_2\cap B_1^c\cap B_2^c)\cup (A_1^c\cap A_2^c\cap B_1)
\\[1ex] & \subseteq (A_1\ca... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1534186",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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Are there any limit questions which are easier to solve using methods other than l'Hopital's Rule? Are there any limit questions which are easier to solve using methods other than l'Hopital's Rule? It seems like for every limit that results in an indeterminate form, you may as well use lHopital's Rule rather than any o... | If you try to use L'Hospital's rule to evaluate
$$ \lim_{x\to\infty} \frac{2x}{x+\sin x} $$
you end up with
$$ \lim_{x\to\infty} \frac{2}{1+\cos x} $$
which spectacularly fails to converge. But the original limit does exist (it is $2$).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1534296",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
"answer_count": 8,
"answer_id": 0
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Coordinates of a Change of Basis Let $\{x_1, x_2, x_3\}$ and $\{y_1, y_2, y_3\}$ be basis of of $\mathbb{R}^3$ with a basis transformation:
$$
y_1 = 2x_1 - x_2 - x_3
$$
$$
y_2 = -x_2
$$
$$
y_3 = 2x_2 + x_3
$$
What are all the vectors that have the same coordinates with respect to the two bases?
| All you really want to do it find the values of $x_1,x_2,x_3$ such that you get back $x_1,x_2,x_3$ when you change the basis. So $y_1=x_2$, $y_2=x_2$ and $y_3=x_3$ which you then put into your system of equations:
$$ x_1=2x_1-x_2-x_3\\ x_2=-x_2 \\ x_3=2x_2+x_3 $$
So from that we can see $x_2=0$ so then we are just solv... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to find the preimage of a set I'm trying to prove continuity of two different functions $f\colon A\to B$. I know that to find continuity, the inverse of all open subsets in $B$ must be open in $A$. I also know that to find the inverse of an open subset you have to find the preimage. I just don't know how to find th... | Let $f:A\to B$ be a map and let $S$ be a subset of $B$, then the preimage of $S$ under $f$ is
$$f^{-1}(S)=\{a\,|\,f(a)\in S\}.$$
For example if $A=\{1,2,3\}$, $B=\{a,b,c\}$ and $f$ is map defined by $f(1)=f(2)=a$, $f(3)=b$, then
$$f^{-1}(\emptyset)=\emptyset,\\
f^{-1}(\{a\})=\{1,2\},\
f^{-1}(\{b\})=\{3\},\
f^{-1}(\{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1534477",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Energy method for one dimensional wave equation with Robin boundary condition Show that the initial-boundary value problem
\begin{align}
& {{u}_{tt}}={{u}_{xx}}\text{ }(x,t)\in \left( 0,l \right)\times \left( 0,T \right),\text{ }T,l>0 \\
& u\left( x,0 \right)=0,\text{ }x\in \left[ 0,l \right] \\
& {{u}_{x... | $\newcommand{\e}{\epsilon}$The first $(?)$ is indeed questionable. Instead, note that you have shown that
$$E'(t) = -u_{t}(l, t)u(l, t) -u_{t}(0, t)u(0, t). \tag{$1$}$$
Defining $\e : [0, \infty) \to \Bbb R$ by $$\e(t) := \frac{1}{2}[u^{2}(l, t) + u^{2}(0, t)],$$
we see that $(1)$ tells us that $(E + \e)' \equiv 0$.
Mo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1534596",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
solving the limit of $(e^{2x}+x)^{1/x}$ I tried to solve for the following limit:
$$\lim_{x\rightarrow \infty} (e^{2x}+x)^{1/x}$$
and I reached to the indeterminate form:
$${4e^{2x}}\over {4e^{2x}}$$
if I plug in, I will get another indeterminate form!
| I have no idea how you got $\lim\limits_{x\to\infty}\frac{4e^{2x}}{4e^{2x}}$, but as has been pointed out, that limit is easily evaluated: the fraction is identically $1$, so the limit is also $1$.
Let $L=\lim\limits_{x\to\infty}\left(e^{2x}+x\right)^{1/x}$; then
$$\ln L=\ln\lim_{x\to\infty}\left(e^{2x}+x\right)^{1/x}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1534694",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 1
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Is it correct to think of the Laplacian as the divergence of a gradient field? Factoring out the notation, I see that
$$\nabla^2(\phi) = \nabla \cdot \nabla(\phi) = \nabla \cdot (\nabla(\phi)) $$
which looks something like the divergence of the gradient of phi.
Is it actually true?
Thanks,
| Yes, it is correct. Indeed, let $\phi \colon \mathbb{R}^N \to \mathbb{R}$ be of class $C^2$, then
\begin{align}
\operatorname{div}(\nabla\phi) := \sum_{i = 1}^N\frac{\partial}{\partial x_i}(\nabla\phi)_i = \sum_{i = 1}^N\frac{\partial^2 \phi}{\partial x_i^2} =: \Delta\phi.
\end{align}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1534757",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Who determines if a mathematical proof is valid? I'm studying mathematics and as you all know the most important things in mathematics are proofs.
My question is, who determines if a proof that someone invents in mathematics is valid? Is there some mathematics professors who check all people's proofs in the world?
If ... | That's a very interesting question. Part of the answer is already included in the question itself - for it is not evident at all that the question should not start with the word "what".
As was pointed out -- correctly I believe -- a determination of the validity of a mathematical proof is a social process. This may com... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1534848",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Evaluate $\lim_\limits{x \to 0}\frac{x}{\sqrt[n]{1+ax} \cdot \sqrt[k]{1+bx} -1}$ For all $n,k \in N a,b > 0$
$$\lim_{x \to 0}\frac{x}{\sqrt[n]{1+ax} \cdot \sqrt[k]{1+bx} -1} = \lim_{x \to 0}\frac{x}{(1+ \frac{ax}{n})(1+ \frac{bx}{k})- 1}= \lim_{x \to 0}\frac{x}{x(\frac{a}{n} + \frac{b}{k}) + \frac{ab}{nk}x^2} = \lim_{x... | The limit is fine. Perhaps with a view to formalizing the procedure is appropriate to introduce an infinite series , and then Landau notation.
$$O(g(x)) = \left\{\begin{matrix} f(x) : \forall x\ge x_0 >0 , 0\le |f(x)|\le c|g(x)| \end{matrix}\right\}$$
We will also use:
$$f_1=O(g_1)\wedge f_2=O(g_2)\implies f_1f_2=O(g_1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1534927",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Show that $\int_{0}^{+\infty} \frac{\sin(x)}{x(x^2+1)} dx = \frac{\pi}{2}\left(1-\frac{1}{e}\right) $ I'm trying to show that
$$\int_{0}^{+\infty} \frac{\sin(x)}{x(x^2+1)} dx = \frac{\pi}{2}\left(1-\frac{1}{e}\right) $$
using Jordan's lemma and contour integration.
MY ATTEMPT: The function in the integrand is even, ... | Worked this out, it's a combination of both methods I mention. You need to go around the contour $C_r+C_e+(-R,-e)+(R,e)$. By the only theorem on section 4.3 of Ablowitz & Fokas the integral around $C_e$ will go to $i\pi$. By the approach I mentioned in the first part of my attempt, the integral around $C_R$ will go to ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1535013",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 2
} |
Why does the principle value of $z^{\frac{1}{2}} := \sqrt{r}e^{i\frac{\theta}{2}}$ need to be restricted to one rotation in order to be continuous? I understand that the function $f(z) = z^{\frac{1}{2}}$ is multi-valued, since it has exactly two solutions for every $z$. However if I take the principle value square root... | $f(r,\theta)$ is continuous, but the map $z\mapsto(r,\theta)$ is multi-valued, due to $Arg(z)$ is multivalued. If you want $z\mapsto(r,\theta)$ to be continuous, you have to select a single valued branch and restrict the range of $\theta$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1535120",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Hahn-Banach separation theorem with a countable subset of functionals For a separable Banach space $X$, the unit sphere of $X^*$ always contains a countable set $D$ such that
$$ \left\Vert x \right\Vert = \sup_{f \in D} \left\vert f(x) \right\vert \qquad \mbox{ for every }x\in X.$$
Assume that $A\subset X$ is nonempty,... | Not in general. Here is a counterexample.
Let $X = C([-1,1])$. For $t \in [-1,1]$, let $\delta_t$ denote the point mass / evaluation functional $\delta_t(x) = x(t)$. Let $D = \{ \delta_q : q \in [-1,1] \cap \mathbb{Q}\}$. Then $D$ is countable and we have $\|x\| = \sup_{f \in D} |f(x)|$ for every $x \in X$.
Let $$y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1535212",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Prove $\lim_\limits{n \to \infty}-\cos(\pi\sqrt{4n^2+10})$ exists and is equal to $-1$ My idea is to take two subsequence converging to different limits. On this step I have problem. How can I set equation of subsequence when $\cos$ has certain value and $n \to \infty$?
$$
\lim_{n \to \infty}-\cos(\pi\sqrt{4n^2+10})
$... | HINT:
The cosine is continuous and $$\pi\sqrt{4n^2+10}= 2\pi n\left(1+\frac5{4n^2}+O\left(\frac{1}{n^4}\right)\right)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1535280",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Problem on entire function which is reduced to constant Suppose $f$ is an entire function satisfying any one of the following conditions for all $z\in \mathbb C$
(1) im$f(z)$ has no zeros
(2)$|f(z)|\geq 1$.
Then f is constant.
My thought: For(2) since $|f(z)|\geq 1$ ,then $f$ has no zero in $\mathbb C$.
Define $g=1... | Hints: 1) Show that ${\rm Im}(f(z))>0$ for all $z$, or ${\rm Im}(f(z))<0$ for all $z$; (if ${\rm Im}(f(z_1))>0$ and ${\rm Im}(f(z_2))<0$, put $g(t)={\rm Im}(f((1-t)z_1+tz_2))$ for $t\in [0,1]$; $g$ is clearly continuous.)
2) If you are in the first case, and if $h(z)=f(z)+i$, what can you say of $|h(z)|$ ?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1535383",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
The set of traces of orthogonal matrices is compact
Is the following set compact: $$M = \{ \operatorname{Tr}(A) : A \in M(n,\mathbb R) \text{ is orthogonal}\}$$ where $\operatorname{Tr}(A) $ denotes the trace of $A$?
In order to be compact $M$ has to be closed and bounded.
$\|A\|=\sqrt {\sum_{i,j} {a_{ij}}^2}=\sqrt n... | Hint: To show that $A$ is orthogonal, consider $\lim_{n \to \infty} \left\| A_n^TA_n - I \right\|$, noting that the function
$$
f(X) = \left\| X^TX - I \right\|
$$
is continuous.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1535494",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 2
} |
What does it mean to represent elements of an ideal? Say I have the polynomial $x^9 + 1$
Then: $x^9 + 1 = (x+1)(x^2 + x + 1)(x^6 + x^3 + 1)$
is a complete factorization over $GF(2)$ of $x^9 + 1$
The dimension of each ideal is: length $n - deg(ideal)$
So for $n=9$, dimension of $(x+1)$ = $9-1=8$
of $(x^2 + x + 1) = 9 -... | Let’s call $f=x^6+x^3+1$. You want three linearly independent elements of the ideal $(f)$ of the ring $R=\Bbb F_2[x]/(x^9+1)$. Since $(f)$ is just the set of multiples of $f$, you certainly have $1\cdot f$, $xf$, and $x^2f$. Notice that $x^3f=x^9+x^6+x^3=1+x^6+x^3=1\cdot f$, already counted. I’ll leave it to you to sho... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1535620",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Matrix exponential of non diagonalizable matrix? I am currently self-learning about matrix exponential and found that determining the matrix of a diagonalizable matrix is pretty straight forward :).
I do not, however, know how to find the exponential matrix of a non-diagonalizable matrix.
For example, consider the matr... | Laplace Transforms approach is an useful for finding $e^{A t}$, whether $A$ is diagonalizable or non-diagonalizable. It is a very useful method in Electrical
Engineering.
Basically, we use the formula
$$
e^{A t} = \mathcal{L}^{-1}\left[ (s I - A)^{-1} \right]
$$
As an illustration, consider the defective matrix
$$
A = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1535731",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 3,
"answer_id": 2
} |
Evaluate $\lim_{x \to 0} \left(\frac{ \sin x }{x} \right)^{\frac{1}{x^2}}$ $$\lim_{x \to 0} \bigg(\frac{\sin x}{x} \bigg)^{\frac{1}{x^2}}$$
The task should be solved by using Maclaurin series so I did some kind of asymptotic simplification
$$\lim_{x \to 0} \bigg(\frac{\sin x}{x} \bigg)^{\frac{1}{x^2}} \approx \lim_{x \... | $$
\lim_{x \to 0} \left( 1 - \frac {x^2}6\right)^{\frac 1{x^2}} = \lim_{x \to 0} \left [ \left( 1 - \frac {x^2}6\right)^{\frac 6{x^2}} \right ]^{\frac 16} = \left [ \left (e^{-1} \right ) \right ]^{\frac 16} = e^{-\frac 16}
$$
Here, I used somewhat modified limit regarding Euler's number
$$
\lim_{t \to 0} \left( 1 - t\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1535827",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Can I think of the conformal mapping w = (z+1/z) as a linear fractional transformation? The mapping
$$w = z + \frac{1}{z}$$
looks linear in $z$. However, it would not be in the form
$$\frac{Az+B}{Cz+D}$$
since putting the two terms together gives
$$\frac{z^2+1}{z}$$
So my question is: is this mapping a linear frac... | No: linear fractional transformations are bijective, and this map isn't: consider $z=2$ and $z=1/2$.
You can take a look at the graph here: http://davidbau.com/conformal/#z%2B1%2Fz
There is some nice symmetry in the fact that $f(z)=f(1/z)$, so it's "symmetric about the unit circle" (up to flipping across the real axis)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1535898",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove Piecewise Function Integrable $$
f(x) =
\begin{cases}
-2, & \text{if }x < 0 \\
1, & \text{if }x > 0\\
0, & \text{if }x = 0
\end{cases}
$$
Hey guys I need some help showing that this function is integrable on the closed interval $[-1,2]$.
So far my idea has been to show $$U(f,P)-L(f,P) < \epsilon$$ for some $\eps... | Notice that for any $x$ at which $f$ is discontinuous, $x$ can be contained in an interval as small as you please. Make sure to include this small interval in your partition.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1535997",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Reducing a 2nd order system of ODEs to a 1st order system I need to numerically solve the following system of ODEs:
$$x''(t)=- \frac{3x}{(x^2+y^2)^{3/2}}$$
$$y''(t)=- \frac{3y}{(x^2+y^2)^{3/2}}$$
I know that I must convert to a system of 1st order equations but I am having trouble doing so...Could someone provide me wi... | All that needs to be done to convert to a system of first order ODEs is to play some games with your notation, in this case it's particularly simple because you don't have any first derivatives.
So let's say we choose to define
\begin{align}
x_1 &\equiv x, \\
x_2 &\equiv x_1', \\
y_1 &\equiv y, \\
y_2 &\equiv y_1'.
\en... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1536071",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How can I derive monotonically increasing polynomial functions with steep curves from [0,1] to [0,1]? For computer graphics reasons, I need a Taylor polynomial function on the interval [0,1] like the classic smoothstep function 3x^2-2x^3 which outputs a monotonically increasing value on the interval [0, 1] but that has... | I want a polynomial, probably low-degree.
So let's try a + bx + cx^2 + dx^3 + e^4.
I want the point at zero to be zero so a must be zero.
I want the point at one to be one so c + d + e must be one.
I want the start to be convex and curve up so the first derivative at zero must be zero and hence b must be zero.
I want t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1536175",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Solve the above program Consider the problem of covering the triangle with vertices at the points
$(0, 0), (0, 1),$ and $(1, 0)$ with a ball of smallest radius.
$$\min r$$
$$s. t. \> x ^2 + y ^2 ≤ r$$
$$(x − 1)^ 2 + y ^2 ≤ r$$
$$x^ 2 + (y − 1)^ 2 ≤ r.$$
Solve the above program
| There’s no need for fancy techniques. Clearly the diameter of the disk must be at least the distance between the points $\langle 0,1\rangle$ and $\langle 1,0\rangle$, which is $\sqrt2$, so let’ see what happens if we try a disk of radius $\frac{\sqrt2}2$ centred midway between those points, at $\left\langle\frac12,\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1536336",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
how to solve this counting problem? Each user on a computer system has a password, which is six to eight characters long, where each character is an uppercase letter or a digit. Each password must contain at least two digits. How many possible passwords are there?
Is the answer: $(36^6 - 26^5) + (36^7 - 26^6) + (36^8 ... | I would use
$$\left(36^6 - 26^6 - \binom{6}{1}\cdot 26^5\cdot 10 \right) + \left(36^7 - 26^7 - \binom{7}{1}\cdot 26^6\cdot 10 \right) + \left(36^8 - 26^8 - \binom{8}{1}\cdot 26^7\cdot 10 \right).$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1536454",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
$z_1,...z_n$ are the $n$ solutions of $z^n =a$ and $a$ is real number, show that $z_1+...+z_n$ is a real number I was actually trying this simple question, might be just me being really rusty not doing maths for a very long time. I tried adding all the $\theta$ up on all zs but it doesn't seem to work.
Anyway as the ti... | Use thae fact that $$z^n -a =z^n -(z_1 +...+z_n ) z^{n-1} +...+ z_1 \cdot z_2 \cdot ...\cdot z_n $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1536532",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Positive equilibria for a system of eqautions I have the following system of equations
\begin{align}
\frac{dx}{d \tau} &= x \left(1-x-\frac{y}{x+b} \right) \\
\frac{dy}{d \tau} &= cy \left(-1+a\frac{x}{x+b} \right)
\end{align}
I am asked to show that if $a<1$, the only nonnegative equilibria are $(0,0), (1,0)$.
So fir... | If $x \ne 0$, We have
$$ 1 - a\frac{x}{x+b} = 0 $$
$$ a\frac{x}{x+b} = 1 $$
$$ ax = x + b$$
$$ (a-1)x = b$$
$$ x = \frac{b}{a-1} $$
If $a < 1$ and $b > 0$, $x < 0$ and therefore is not a non-negative solution
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1536632",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
What is the value of $ x(\log x)$ when $x=0$ and $x\not \to 0$? I know that $ x(\log x)\to 0$ when $x \to 0$, but some people say that since $\log x$ is not defined when $x=0$, so the value of $ x(\log x)$ can also not be found when $x=0$.
But isn't it true that if you multiply anything with 0, the answer comes out ... | Since $\ln(x)$ is not defined, $x\ln(x)$ is not defined. Difficult to multiply things that do not exist.
However, since the existing definitions do not cover the case $\ln(0)$, you could arbitrarily choose a value for $\ln(0)$: $0$, $1$, $e$, $1664$ or your mother's birthday... There is no problem doing that, but there... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1536716",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 2
} |
upper bound $\sum_{i=1}^n a_i (a_i-1)/2$ using a function of $\sum_{i=1}^n a_i$ Given $a_i \in \mathbb{N}\cup\{0\}$ and
define
$$
A(n) = \sum_{i=1}^n a_i (a_i-1)/2
$$
and
$$
B(n) = \sum_{i=1}^n a_i
$$
Any ideas how to upper bound $A(n)$ as a "function" of $B(n)$? (the tighter, the better; form of this "function" doe... | Let $f(x)=\dfrac{x(x-1)}{2}$. We
note first that
$$\forall\,x,y\ge0,\quad f(x+y)-f(x)-f(y)=xy\ge0$$
Thus, by induction, for nonnegative numbers $a_1,\ldots,a_n$ we have
$$\sum_{i=1}^n f(a_i)\le f\left(\sum_{i=1}^na_i\right)$$
Thus,
$$A(n)\le \frac{B(n)(B(n)-1)}{2}$$
With equality if $a_1=\cdots=a_{n-1}=0$ for example.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1536839",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
N marbles puzzle: find the heaviest among them. Suppose there are $N$ marbles and a two-pan balance used to compare the weight of 2 things. All of the marbles weigh the same except for one, which is heavier than all of the others. How would you find the heaviest marble in minimum number of comparisons.This link explain... | Each weighing can reduce the number of possibilities to $1/3$ of that from before, using the same argument as in the example you linked. If you have $3^n$ marbles, then you need at least $n$ weighings to find the heavier marble. If you have $3^n < N \leq 3^{n+1}$ marbles you should need $n+1$ weighings--since $N$ is no... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1536963",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Prove Graph G is in Vizing Class 2 if $\alpha$'(G) < |e(G)|/$\Delta$(G) G is a simple graph with e edges, maximum vertex degree $\Delta$ and edge independence number $\alpha$' which satisfies $\alpha$' < e/$\Delta$.
What does this inequality mean? How is it helpful in proving that G is Vizing Class 2?
I'm struggling to... | I discovered the answer was simpler than I thought.
$\alpha$' < e/$\Delta$ --> $\Delta$ < e/$\alpha$'
$\chi$' >= e/$\alpha$'
$\chi$' > $\Delta$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1537039",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
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