Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
A Taylor series question The Taylor series for $cos(x)$ about $x=0$ is $1-x^2/(2!)+x^4/(4!)-x^6/(6!)+...$
If $h$ is a function such that $h'(x) = cos(x^3)$, then the coefficient of $x^7$ in the Taylor series for $h(x)$ about $x = 0$ is?
| The Taylor expansion of $\cos t$ is
$$1-\frac{t^2}{2!}+\frac{t^4}{4!}-\frac{t^6}{6!}+\cdots.$$
Let us cross our fingers and treat this as a "long" polynomial. Then substituting $x^3$ for $t$, we get
$$\cos(x^3)=1-\frac{x^6}{2!}+\frac{x^{12}}{4!}-\frac{x^{18}}{6!}+\cdots.$$
This is $h'(x)$. Integrate term by term, agai... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/751025",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
What is Cramer's rule used for? Cramer's rule appears in introductory linear algebra courses without comments on its utility. It is a flaw in our system of pedagogy that one learns answers to questions of this kind in courses only if one takes a course on something in which the topic is used.
On the discussion page to... | In control theory, the closely related rule
$$A^{-1} = \frac{1}{\det A} \operatorname{Adj}(A)$$
where $\operatorname{Adj}(A)$ is the adjugate matrix, is used for going from a state space representation to a transfer function description (doing all computations by hand).
Explicitly, if we are given a state space represe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/751089",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 5,
"answer_id": 2
} |
Jumping back into Calculus III At the age of 30 I am going back to school for Electrical Engineering. Because of the way higher education works, all of my previous college coursework is being transferred, which does not allow you to retake classes that were already successfully completed. Since I took Calc I and II an... | Calculus Early Transcendentals by James Stewart, I think it is one of the best books out there. Apart from Khan Academy on Youtube, you have MIT Multivariable Calculus and UCBerkely.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/751154",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 3
} |
Finding an expression for the complex number Z^-1 So I want to find out an expression to express:
$$z^{-1}$$
I know the answer is:
$$z^{-1} = \frac{x-iy}{x^2+y^2}$$
But how would I go about proving this/the steps to this?
| Our intuition from the real numbers tells us that the inverse of a real number $c$ is $1/c$.
Likewise, we conjecture that the inverse of a complex $z \in \mathbb{C}$ is $\frac{1}{z} = \frac{1}{x + iy}$. To get this into a more recognizable form, multiply the numerator and denominator by $z^* = x - iy$. From here, w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/751238",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Which triangular numbers are also squares? I'm reading Stopple's A Primer of Analytic Number Theory:
Exercise 1.1.3: Which triangular numbers are also squares? That is, what conditions on $m$ and $n$ will guarantee that $t_n=s_m$? Show that if this happens, then we have:
$$(2n+1)^2-8m^2=1,$$
a solution to Pell's equat... | As the other answers have explained, the multiplication by 4 is to make things neater. However, on closer look, the formulation
(2n+1)^2 - 1 = 8m^2
doesn't really simplify the situation.
This is because (2n+1), an odd number, when squared, will always be one more than [8 times a triangular number]. This formulation sim... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/751316",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 3
} |
Checking a solution of a PDE I have the following PDE:
\begin{equation}
-yu_x + xu_y = 0 \quad\text{where } u(0, y) = f(y)
\end{equation}
I derived a solution as follows:
\begin{align}
-yu_x + xu_y =& 0 \\
\iff& \nabla u(x,y)\dot \langle -y, x\rangle = 0 \\
\implies& \frac{dy}{dx} = \frac{-x}{y} \\
\iff& ydy ... | Since we conjecture $u(x,y)=f(x^2+y^2)$, let us (carefully) apply calculus and verify $-yu_x+xu_y=0$.
Note (by chain rule),
$$ u_x = f'(x^2+y^2)*2x, \qquad u_y = f'(x^2+y^2) * 2y. $$
Thus, $-yu_x+xu_y=-2xyf'+2xyf'=0$.
So, you had it, except a small detail with the chain rule. Another way to have caught the mistake -- $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/751433",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
The group of rigid motions of the cube is isomorphic to $S_4$. I want to solve the following exercise from Dummit & Foote. My attempt is down below. Is it correct? Thanks!
Show that the group of rigid motions of a cube is isomorphic to $S_4$.
My attempt:
Let us denote the vertices of the cube so that $1,2,3,4,1$ trac... | Surjective: The group of rigid motions of a cube contains $24$ elements, same as $S_4$. Proof - A cube has $6$ sides. If a particular side is facing upward, then there are four possible rotations of the cube that will preserve the upward-facing side. Hence, the order of the group is $6\times 4 = 24$.
Injective: A cube ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/751507",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
} |
Limit of a polynomic-exponential sequence I have to calculate the following limit:
$$L=\lim \limits_{n \to \infty} -(n-n^{n/(1+n)})$$
I get the indeterminate form $\infty - \infty$ and I don't know how to follow. Any idea?
Thank you very much.
| HINT :
$$n - n^{n/1+n} = n - n^{1+1/n} = n(1 - n^{1/n}) = \frac{1 - n^{1/n}}{1/n}$$
Now use L'Hôpital's rule.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/751584",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Showing an endomorphism is not surjective Let $$A=\begin{pmatrix}2&-2\\2&-2\end{pmatrix}$$ and the endomorphism $f_A:M_2(\mathbb R)\longrightarrow M_2(\mathbb R); B\longmapsto AB$. I want to show that $f_A$ is not surjective.
My try: $\ker f_A$ is clearly shown to be containing elements other than the null matrix, so $... | Your approach works (it'd be better to explicitly write down a nonzero element of $\mathrm{ker} \, f_A$). Alternatively, every element of the image satisfies $\det(f_A(B)) = \det(AB) = 0 \cdot \det(B) = 0$. Of course there are matrices with nonzero determinant, and they are therefore not in the image.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/751717",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find $\int_\Gamma\frac{2z+j}{z^3(z^2+1)}\mathrm{d}z$ where $Γ:|z-1-i| = 2$ pls, some ideas for integral solution (residue theory)?
$$\int_\Gamma\dfrac{2z+j}{z^3(z^2+1)}\mathrm{d}z$$
Where $Γ:|z-1-i| = 2$ is positively oriented circle.
Thx, for help!
| So, my solution is:
Is it correct ??
pole z1 = 0 ( order 3 pole );
pole z2 = -i ( simple pole );
pole z3 = i ( simple pole );
z1 : |0-1-i| = sqrt(2) < 2 => in circle;
z2 : |-i-1-i| = sqrt(5) > 2 => not in circle;
z3 : |i-1-i| = 1 < 2 => in circle;
$$
\underset{z=0}{res}\frac{2z+i}{z^3(z^2+1)}=-i
$$
$$
\underset{z=i}{re... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/751953",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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How to calculate this area in $\mathbb{R}^2$? Write the area $D$ as the union of regions. Then, calculate $$\int\int_Rxy\textrm{d}A.$$
First of all I do not get a lot of parameters because they are not defined explicitly (like what is $A$? what is $R$?).
Here is what I did for the first question:
The area $D$ can be wr... | You should write
$$
D =\{(x,y): -1<x<1, -1<y<1+x^2
\}
-
\{ (x,y): 0<x<1, -\sqrt{x}<y<\sqrt{x}
\}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/752045",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
When is a symmetric 2-tensor field globally diagonalizable? Suppose that $\mathbb{R}^n$ has a Riemannian metric $g$. Let $h$ be a smooth symmetric 2-tensor field on $\mathbb{R}^n$.
At any point $p \in \mathbb{R}^n$, there is a basis of $T_p \mathbb{R}^n$ in which $h$ is diagonal. Is it always possible to find a globa... | Once I had similar question. I asked if you are given continuous matrix valued function $A:\Omega \rightarrow \mathbb{R}^{n\times n}$. Can you find continuous matrix valued functions $D$ diagonal and $S$ orthogonal, such that
$$
A(x) = S(x)D(x)S^T(x)
$$
for all $x\in \Omega$ ?
The answer is negative. Take this matrix... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/752115",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Geometry Right triangles in a rectangle, find the area.
Please help, I've been struggling to figure out this problem for too long...
Given the area of rectangle $ABCD = 1200 \text{ unit}^2$, find the area of right triangle $ABE$
| We have $[ABCD]=1200$, therefore the area of $\Delta{ABD}=\dfrac{1}{2}[ABCD]=600$. Now, calculate length $AD$ and $BD$.
$$
\begin{align}
[ABD]&=600\\
\dfrac{1}{2}AB\cdot AD&=600\\
\dfrac{1}{2}\cdot40\cdot AD&=600\\
20\cdot AD&=600\\
AD&=30
\end{align}
$$
Using Phytagoras' formula, we get
$$
BD^2=AB^2+AD^2\quad\Rightarr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/752186",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Are curves closed in $\mathbb{R}\times \mathbb{R}$ with the standard topology? Given the graph of the curve $y=\frac{1}{x}$, can we determine if the curve is closed or open in $\mathbb{R}^{2}$ with the standard topology?
| Perhaps the easiest way to show closedness of
$$G=\{(x,\ 1/x)\mid x\ne 0\}\subset\Bbb R^2$$
is to note that it is the preimage of $\{1\}$ under the map
$$\Bbb R\times\Bbb R→\Bbb R,\qquad (x,y)\mapsto x\cdot y$$
and this map is continuous and $\{1\}$ is closed.
Alternatively, note that $G=G(f)$, the graph of the con... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/752269",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Functions definition + question Am I correct in saying that for Functions, the below is the correct definition:
For each value of x in the domain there is only one value of y in the range.
Hence, the picture below means that it is not a many-to-one function as the values of x do not map onto a single value of y in the ... | A function is a relation between a set of inputs and a set of permissible outputs (determined by the relation), with the property that each input is related to exactly one output. So you need to make sure to change "only" to "exactly" or to " one and only one".
I emphasize one and only one because each input value must... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/752325",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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So-called Artin-Schreier Extension Let $F$ be a field of characteristic $p$. Let $K$ be a cyclic extension of $F$ of degree $p$. Prove that $K=F(\alpha)$ where $\alpha$ is a root of the polynomial $p(x) = x^{p} - x - a$ for $a \in \mathbb{F}$.
I've seen, and attempted, a lot of problems that look similar. But I'm not r... | Let $\sigma$ be a generator of $G_{K/F}$. The equation $x^p-x=a$ can be written in two different ways:
$$\begin{cases}x(\sigma x)\cdots(\sigma^{p-1}x) & =a \\ x(x+1)\cdots(x+p-1) & =a\end{cases}$$
The first follows because $a$ is the opposite of the constant term of $x$'s minimal polynomial thus giving its norm (as ei... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/752410",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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A bounded integral I want to show that there exists $K\in\mathbb{R}^+$ such that
$$\left|\int_{1}^x \sin(t+t^7)dt \right|<K$$
for all $x\ge 1$. Intuitively, I'm quite sure it is true, but I can't find a formal proof. Any idea?
| Let $f(x) = x + x^7$, and $g(t)$ its inverse function on $[0,\infty)$.
Then $$\int_0^x \sin(x + x^7)\ dx = \int_0^{f(x)} \sin(t) g'(t)\ dt$$
It can be shown that as $t \to \infty$,
$$g(t) = t^{1/7} - \dfrac{1}{7} t^{-5/7} + O(t^{-11/7})$$
and
$$g'(t) = \dfrac{1}{7 g(t)^6 + 1} = \dfrac{1}{7} t^{-6/7} + O(t^{-12/7})$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/752606",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Jacobian Linearisation of Non Linear System Can any one please solve the below problem. It is related to Jacobian Linearisation of Non Linear System
I have only got till here
| You can try something like this:
$$ \dfrac{dx_1}{dt}=f_1(u,x_1,x_2)$$
$$ \dfrac{dx_1}{dt}=f_2(u,x_1,x_2)$$
$$ \dfrac{d\Delta x_1}{dt}=(\dfrac{df_1}{du})_0\Delta u + (\dfrac{df_1}{d x_1})_0\Delta x_1 +(\dfrac{df_1}{dx_2})_0\Delta x_2 $$
$$ \dfrac{d\Delta x_2}{dt}=(\dfrac{df_2}{du})_0\Delta u + (\dfrac{df_2}{d x_1})_0\D... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/752721",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Why isn't the zero after the decimal in $0.01$ significant? Why isn't the zero after the decimal in $0.01$ significant? Although it is pretty obvious that the zero before the decimal is insignificant, I don't understand why the zero after the decimal is not significant.
| Significant figures are used to denote the precision of a measurement. The leading zeros are not significant because they don't give us information about the precision of the measurement.
Let's say you measure something with a meter stick that only has centimeter markings (no millimeters). You get that the object is $8... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/752884",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Proving $\gcd \left(\frac{a}{\gcd (a,b)},\frac{b}{\gcd (a,b)}\right)=1$ How would you go about proving that $$\gcd \left(\frac{a}{\gcd (a,b)},\frac{b}{\gcd (a,b)}\right)=1$$
for any two integers $a$ and $b$?
Intuitively it is true because when you divide $a$ and $b$ by $\gcd(a,b)$ you cancel out any common factors betw... | Assume WLOG that $a, b \geq 1$. Let $m = \dfrac{a}{\gcd(a,b)}$, and $n = \dfrac{b}{\gcd(a,b)}$, and let $c = \gcd(m,n)$. Then $c \mid m$, and $c \mid n$. This means: $(c\cdot \gcd(a,b)) \mid a$, and $(c\cdot \gcd(a,b)) \mid b$. So $(c\cdot \gcd(a,b)) \mid \gcd(a,b)$. but $\gcd(a,b) \mid (c\cdot \gcd(a,b))$. Thus: $c\cd... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/752928",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 9,
"answer_id": 8
} |
Finitely many prime ideals lying over $\mathfrak{p}$
Let $A$ be a commutative ring with identity and $B$ a finitely generated $A$-algebra that is integral over $A$. If $\mathfrak{p}$ is a prime ideal of $A$, there are only finitely many prime ideals $P$ of $B$ such that $P\cap A=\mathfrak{p}$.
Let me say that I am a... | One very useful tool is:
For a ring map $A\to B$, and $\mathfrak{p}$ a prime ideal of $A$, the prime ideals which contract to $\mathfrak{p}$ are in $1:1$ correspondence to the prime ideals of $\kappa(\mathfrak{p})\otimes_AB$, where $\kappa(p)=Q(A/\mathfrak{p})$ is the quotient field of the domain $A/\mathfrak{p}$.
N... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/753042",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 2,
"answer_id": 0
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Set of solutions for a binomial inequality I bumped into the following inequality:
$${a-b\choose c}{a\choose c}^{-1} \le \exp\left(-\frac{bc}{a}\right)$$
Playing with it a little bit, trying to bound it asymptotically for large $a$'s, using Stirling's approximation, I ended up with nothing. Finally I decided to put som... | Actually, I don't know if you want LHS $\ge $ RHS or LHS $\le $ RHS.
But here is an example: If you take $a=b+1=c+2$ --- e.g. $a=5,b=4,c=3$, then LHS $=0$ and the RHS is positive. So LHS $<$ RHS.
I think that given your restrictions $a>b>c$, it can never hold that LHS $\ge $ RHS.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/753146",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Sum of fractions of squared sines I'm trying to prove the following approximate identity for $p$ integer:
$$
\sum_{l=1}^m\frac{\sin^2\left(\frac{\pi l}{p}\right)}{\sin^2\left(\frac{\pi l}{mp}\right)}\sim \frac{m^2(p-1)}{2}+O(m)
$$
Things I have tried:
*
*Convert to an integral through a Riemann sum, however, the fun... | For large $m$, the quantity $\pi \ell/(m p)$ is small except where $\ell \approx m$. Even then, the argument of the sine is small for even moderate values of $p$, so to first order we can replace the sine by its argument. Thus the ratio looks like, approximately,
$$\left (\frac{m p}{\pi} \right )^2 \sum_{\ell=1}^m \f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/753276",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Centralizer of $SO(n)$ Given the set $M(n,\mathbb C)$ of all complex $n\times n$ matrices, what's the centralizer of $SO(n)$ in $M(n,\mathbb C)$?
For $n=2$, the centralizer must be the matrices $A$ such that $RA=AR$ where $R$ is a rotation matrix. Since 2D rotations commute, I can see $A$ is probably a rotation matrix ... | If $A$ centralises $SO(n)$, it commutes with every $R$ of the form $R=P\left[\pmatrix{0&-1\\ 1&0}\oplus I_{n-2}\right]P^T$ where $P$ is a permutation matrix. Therefore, when $n\ge3$, $A$ must be a diagonal matrix. Consider the equality $AR=RA$ again, we can further infer that $A$ is a scalar multiple of $I_n$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/753373",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Find an $n\times n$ integer matrix with determinant 1 and $n$ distinct eigenvalues Pretty much what the title suggests: for any positive integer $n$, I'm looking for an $n$-by-$n$ matrix with integer entries, determinant $1$ and $n$ eigenvalues.
In case it is absolutely useless to come up with such a matrix, I'm lookin... | Take any representation of degree $n$ of any symmetric group $S_m$. All the matrix entries will be integers but determinant could be $\pm 1$. As suggested by others, we can change all the signs in the first row, if needed, and get integer matrices of determinant $+1$. As all matrices are of finite order, they will ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/753485",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Proving that the line integral $\int_{\gamma_{2}} e^{ix^2}\:\mathrm{d}x$ tends to zero Let $f(z) = e^{iz^2}$ and $\gamma_2 = \{ z : z = Re^{i\theta}, 0 \leq \theta \leq \frac{\pi}{4} \} $. All the sources I have found online, says that the line integral
$$ \left| \int_{\gamma_2} e^{iz^2}\mathrm{d}z \right| $$
tends to... | You can do this as follows. Start as you did with
$$\left\vert\int_{\gamma_2} e^{iz^2}dz\right\vert\leq R\,\int_0^{\frac\pi4} e^{-R^2\sin 2\theta} d\theta\, . $$
Then observe that on $[0\frac\pi4]$ you have
$$\sin 2\theta \geq \frac4\pi \, \theta\, ,$$
thanks to the concavity of the sine function on $[0\frac\pi2]$.
It ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/753676",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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Inequality with two binomial coefficients I am having trouble seeing why
$$
\binom{k}{2} + \binom{n - k}{2} \le \binom{1}{2} + \binom{n - 1}{2} = \binom{n - 1}{2}
$$
| Assuming $1\le n$ and $0\le k\le n$ it's equivalent to
\begin{align*}k(k-1)+(n-k)(n-k-1)&\le(n-1)(n-2)\\
(n-k)(n-k-1)&\le(n-k-1+k)(n-2)-k(k-1)\\
(n-k)(n-k-1)&\le(n-k-1)(n-2)+k(n-2)-k(k-1)\\
0&\le(n-k-1)(k-2)+k(n-k-1)\\
0&\le(n-k-1)(k-1)\end{align*}
So it holds for all $0<k<n$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/753799",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Angle in figure consisting of a square surrounded by semi circles I'd like to know how to get the angle in the following problem:
It is a square with side equal to 1. The radius of each semi circle is equal to the side of the square. How can this angle be determined?
| It's $30^\circ$. Let the point of intersection of "upper" arcs $BD$ and $AC$ be called $E$, and of upper $BD$ with lower arc $AC$ be called $F$. You should recognize that $\triangle ABE$ is equilateral (why?). What about $\triangle ADF$? Now finish.
| {
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"timestamp": "2023-03-29T00:00:00",
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What is a zero morphism in an abelian category I am trying to familiarize myself with some basic category theory
and I am getting confused with what a $0$-morphism is.
If we are in category of say $k$-vector spaces
then I am guessing $0$-morphism would be the map that sends everything to $0$.
In these examples it make... | The zero morphism $A \to B$ can be factored into
$$ A \to 0 \to B $$
where $0$ is a zero object. (i.e. it is a terminal object and an initial object)
As an aside, when you wrote it as "$0$-morphism", my first reaction was that you were referring to the concept from higher category theory; e.g. in $\mathbf{Cat}$, catego... | {
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Show that there is no integer n with $\phi(n)$ = 14 I did the following proof and I was wondering if its valid. It feels wrong because I didn't actually test the case when purportedly $n$ is not prime, but please feel free to correct me.
Assume there exists $n$ such that $\phi(n) = 14$. Assume $n$ is prime. Then $\phi(... | This does not hold. What kind of similar argument are you then talking about?
You can use the fact that $\phi$ is multiplicative. Assume $n = p_1^{a_1}p_2^{a_2}...p_t^{a_t}$, then $\phi(p_1^{a_1}p_2^{a_2}...p_t^{a_t})$ $\phi(p_1^{a_1})\phi(p_2^{a_2})...\phi(p_t^{a_t}) = 2 \cdot 7 = 1 \cdot 14$
Use this to arrive at a c... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Triangles Numbers counting
How measure to number of triangles??? any helps??
I want to calculate it by using a formula.
| My answer has mathematics and observation.
My approach is to get the number of triangles of a paricular size, and then add that to the next size, and so on.
First, as your figure has four small triangles on each side, I take t as $4$. But that comes later.
Starting with triangles of Size $1$. The number of Size $1$ is... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving minimum vertex cover Every vertex cover of a graph contains a minimum vertex cover.
I know the statement to be true but how do I go proving it?
| Let $C = \{v_{1}, ..., v_{k} \}$ be a vertex cover. By definition of a vertex cover, every edge is incident to some vertex in $C$. Suppose $C$ contains no minimum vertex cover. So begin removing vertices inductively. Since there is no minimum vertex cover, we can keep removing vertices from $C$ in this manner. The proc... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that this triangle is equilateral? Given $\triangle ABC$. Let $D$ be the point where the altitude form the $A$ vertex intersect $\overline{BC}$ and the point $E$ is the intersect between the bisector of $\angle ABC$ with $\overline{AC}$. Let $P$ be the point of intersect of $\overline{AD}$ with $\overline{BE}$.... | By the Angle Bisector Theorem in $\triangle ABD$,
$$\frac{|BA|}{|BD|} = \frac{|PA|}{|PD|} = \frac{2}{1}$$
Therefore, $\triangle ABD$ is a $30^\circ$-$60^\circ$-$90^\circ$ triangle; and, then, so is $\triangle BPD$. This implies that your single-tick-mark segments are congruent to your double-tick-mark segments, so tha... | {
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"source": "stackexchange",
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Application of Kodaira Embedding Theorem I am going to give a talk on Kahler manifold. In particular, I will outline a proof of the Kadaira Embedding theorem.
I also wish to give some applications of the theorem. One of the application would be the Riemann bilinear relation on complex torus.
I am searching for other ... | Let me mention some important theorems about Kodaira embedding theorem
Let $X$ be a compact complex manifold, and $L$ be a holomorphic line bundle over $X$ equipped with a smooth Hermitian metric $h$ whose curvature form (locally given by $−i2π∂\bar ∂\log h$) is a positive definite real $(1,1)$-form, and so defines a ... | {
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"source": "stackexchange",
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valuation ring is a field?
suppose $a$ and $a'$ are units of $B$ ,$b$ and $b'$ are the elements of any ideal of $B$. $x$ is a element of $K$.
$K$ consist of $a/a,a/b,b/a,b/b$
$\color{green} x=a/a' \Rightarrow x\in B~and~x^{-1}\in B$
$\color{green} x=a/b \Rightarrow x^{-1}\in B $
$\color{green} x=b/a \Rightarrow x\in ... | I don't follow your reasoning. I think you may be arguing that $K$ has no (nonzero) ideals.
Here is a simple nondegenerate example: let $B$ the the ring of all rational numbers with odd denominator. It is a valuation ring of $\mathbf{Q}$.
$B$ is also a local ring, whose maximal ideal is the one generated by $2$; it is ... | {
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Problem with trigonometric equation I am having trouble solving this equation
$$4\cdot \sin \theta + 2 \cdot \sin 2\theta =5$$
Thank you for your help.
| If you put $t=\tan \frac {\theta}2$ you obtain $$4\cdot\sin \theta+4\cdot \sin \theta\cdot\cos\theta=4\left(\frac {2t}{1+t^2}\right)\left(1+\frac{1-t^2}{1+t^2}\right)=5$$
Multiply though by $(1+t^2)^2$ to obtain $$16t=5(1+t^2)^2$$From which it is clear that any solution has $t$ positive (rhs is positive), and a quick s... | {
"language": "en",
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Find bases of matrix without multiplying This question is related to a solved problem in Gilbert Strang's 'Introduction to Linear Algebra'(Chapter 3,Question 3.6A, Page 190).
Q) Find bases and dimensions for all four fundamental subspaces of A if you know that
$A = \begin{bmatrix}1 & 0 & 0\\2 & 1 & 0 \\ 5 & 0 & 1\end{... | If you think of the product $E^{-1}R$ as the composition of linear applications, then $E^{-1}$ acts "last" and its columns determine somehow the image of $E^{-1}R$ (of course, it also depends on $R$).
More generally, for any functions from any sets that you can compose (assuming $Im(g) \subset D(f)$ where $D(f)$ is the... | {
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How do I solve this definite integral: $\int_0^{2\pi} \frac{dx}{\sin^{4}x + \cos^{4}x}$? $$\int_0^{2\pi} \frac{dx}{\sin^{4}x + \cos^{4}x}$$
I have already solved the indefinite integral by transforming $\sin^{4}x + \cos^{4}x$ as follows:
$\sin^{4}x + \cos^{4}x = (\sin^{2}x + \cos^{2}x)^{2} - 2\cdot\sin^{2}x\cdot\cos^{2... | \begin{aligned}
& \int_{0}^{2 \pi} \frac{d x}{\sin ^{4} x+\cos ^{4} x} \\
=& \int_{0}^{2 \pi} \frac{d x}{\left(\sin ^{2} x+\cos ^{2} x\right)^{2}-2 \sin ^{2} x \cos ^{2} x} \\
=& \int_{0}^{2 \pi} \frac{d x}{1-\frac{\sin ^{2} 2 x}{2}} \\
=& 16 \int_{0}^{\frac{\pi}{4}} \frac{d x}{1+\cos ^{2} 2 x} \\
=& 16 \int_{0}^{\frac... | {
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concerning a cheque A man went into a bank to cash a check. In handling over the money the cashier, by mistake, gave him dollars for cents and cents for dollars. He pocketed the money without examining it and on the way home he spent a nickel. Later on examining it, he found that he had twice the amount of money writte... | Let $D$ be the check's actual number of dollars and $C$ be its actual number of cents. Then the actual amount of the check, expressed in pennies, is
$$A=100D+C$$
The amount the man is given is
$$G=100C+D$$
The pertinent equation is
$$G-5=2A$$
Can you take it from there?
Added later: A couple of people correc... | {
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Prove that this function is injective I need to prove that this function is injective:
$$f: \mathbb{N} \times \mathbb{N} \to \mathbb{N}$$
$$f: (x, y) \to (2y-1)(2^{x-1})$$
Sadly, I'm stumbling over the algebra. Here is what I have so far:
Suppose $f(x, y) = f(a, b)$.
We want to show that $x = a$ and $y = b$.
$$(2y-1)(... | $$(2y-1)2^{x-1}= (2b-1)2^{a-1}$$
If $x\ne a$ then either $x>a$ or $x<a$. Just call whichever one is bigger $a$, so that $x<a$.
Divide both sides by $2$, and repeat $x-1$ times. For example, say $(2y-1)2^{x-1}=1344$. Dividing by $2$ gives $672$; dividing by $2$ again gives $336$; dividing by $2$ again gives $168$; div... | {
"language": "en",
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"source": "stackexchange",
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The derivative of $\rho e^{it}$ Why is
$${df \over dz} \rho e^{it} = i \rho e^{it} \text{?}$$
The product rule states that
$$
{df\over dz}(f_1 \cdot f_2) = f_1 f'_2 + f'_1 \cdot f_2
$$
so why doesn't this imply that
$$
{df\over dz}(\rho \cdot e^{it}) = \rho e^{it} + 0 \cdot e^{it} \text{?} = \rho e^{it} \ne i \rho e^{... | You're confusing three different things with each other:
$$
\frac{df}{dz}, \qquad \frac{d}{dz}, \qquad\frac{d}{dt}
$$
If you had written
$$
\frac{d}{dt} \rho e^{it} = \rho ie^{it}
$$
then it would be correct, but what you have written is at best a misunderstanding of notation.
Applying the product rule, one gets
$$
\be... | {
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Give an equational proof $ \vdash (p \lor \lnot r) \rightarrow (p \lor q) \equiv \lnot q \rightarrow (r \lor p)$ Give an equational proof $$ \vdash (p \lor \lnot r) \rightarrow (p \lor q) \equiv \lnot q \rightarrow (r \lor p)$$
What I tried
$(p \lor \lnot r) \rightarrow (p \lor q)$
Applying De morgan
$\lnot(\lnot ... | Here is a shorter and more 'documented' proof, compared to the original answer to this old question.$
\newcommand{\calc}{\begin{align} \quad &}
\newcommand{\op}[1]{\\ #1 \quad & \quad \unicode{x201c}}
\newcommand{\hints}[1]{\mbox{#1} \\ \quad & \quad \phantom{\unicode{x201c}} }
\newcommand{\hint}[1]{\mbox{#1} \unicode{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/755170",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Why does a positive definite matrix with a repeated eigenvalue have infinitely many square roots? So, if we consider a positive definite matrix $A$, (meaning that $A$ is self-adjoint $(Ax,x) > 0$ and also that $A$ has strictly positive eigenvalues) we see right away that since it is self adjoint, and has an orthonormal... | In case you mean "square root" as $C C^T = A:$
Suppose $A = \lambda^2 I$ two by two,
$$ C \; = \; \lambda
\left( \begin{array}{rr}
\cos \theta & \sin \theta \\
-\sin \theta & \cos \theta
\end{array}
\right) ,
$$
which is to say that a rotated basis of a two-dimensional eigenspace is being used... | {
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Which means adjoint problem of a differential equation? I wanted to know if anyone can help me with the following problem:
Get the adjoint problem (differential equation and boundary conditions) for the problem given by:
$$\frac{d^2 u}{dx^2}=f(x)$$ $$0<x<1$$ $$u(0)=\frac{du}{dx}(0)=0$$
Actually I do not know to be the ... | You need to find an operator adjoint to the given one
$$
L = \frac {d^2}{dx^2}
$$
Condition on real adjoint operator is
$$
\left \langle Lu, v\right \rangle = \left \langle u, L^*v\right \rangle \Longleftrightarrow \int_0^1 \left(Lu\right ) v\ dx = \int_0^1 u \left ( L^* v\right) dx
$$
Now, just do the integration by ... | {
"language": "en",
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Is the reasoning/algebra for my proof correct? (musical tuning theory proof) This isn't for a class, I was just wondering if I would be able to work out a proof for something like this myself for fun, and wanted to verify that my methods are correct. Basically, what I'm trying to prove, in terms of music theory is: Pro... | Yes, your proof is correct.
Here's another proof:
$$\left(\dfrac{3}{2}\right)^m=\left(\dfrac{1}{2}\right)^n\\
\implies 3^m=2^{m-n}\\
\implies m-n=m\log_23\not\in\mathbb{Z}^+\text{ as $m$ is an integer.}\\
\implies n=m(1-\log_23)\not\in\mathbb{Z}^+\text{ as $m$ is an integer.}\\
\implies \{n,m\}\not\subset\mathbb{Z}^+$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/755462",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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fourier series analysis, show that for every integer n, using euler's formulas relating trigonometric and exponential functions Show that for every integer $n$,
$$\int_0^{\pi} \cos nt~\sin t~\mathrm{d}t = \begin{cases} \dfrac{2}{1-n^2} & \text{if } n \text{ is even} \\[10pt]
0 &\text{if } n \text{ is odd} \end{cases}$$... | Consider if the function is even or odd for $n$ even or odd (with respect to the mid point $\pi /2$), this will get your odd case dealt with. To deal with the even case, use the double angle formula repeatedly.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/755581",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Wronskian Bessel Equations I need to compute the wronskian of $J_n$ and $Y_n$ (the Bessel functions of the first and second kinds). I've been able to find in many sources that it is
$$W(J_n,Y_n)=\frac{\pi}{2x}$$, but I haven't been able to prove it. I already could use Abel's formula to get $$ W(J_n,Y_n)=\frac{c}{x}$$... | Hint: Bessel functions of all kinds satisfy the following recurrences:
$$\frac{2n}{x} R_n(x) = R_{n-1}(x) + R_{n+1}(x)$$
$$2\frac{dR_n}{dx} = R_{n-1}(x) - R_{n+1}(x),$$
where $R_n$ can be $Y_n$ or $J_n$.
| {
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"timestamp": "2023-03-29T00:00:00",
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Take 2: When/Why are these equal? This didn't go right the first time, so I'm going to drastically rephrase the query.
As per this previous question, I am wondering if the two series $$\frac{f(a)+f(b)}{2}\frac{(b-a)}{1!}+\frac{f'(a)-f'(b)}{2}\frac{(b-a)^2}{2!}+\frac{f''(a)+f''(b)}{2}\frac{(b-a)^3}{3!}+\cdots$$
and $$\... | Denote a primitive of $f$ by $F$ and add ${1\over2}\bigl(F(a)-F(b)\bigr)$ to both series. Then the first series becomes
$$\eqalign{{1\over2}\sum_{k=0}^\infty \bigl(F^{(k)}(a)-(-1)^kF^{(k)}(b)\bigr){(b-a)^k\over k!}&={1\over2}\sum_{k=0}^\infty F^{(k)}(a){(b-a)^k\over k!}
-{1\over2}\sum_{k=0}^\infty F^{(k)}(b){(a-b)^k\o... | {
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"timestamp": "2023-03-29T00:00:00",
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A question on basis of vectorspaces and subspaces Let $V$ be a finite dimensional vector space and $W$ be any subspace . It is known that if $A$ is any basis of $W$ then by "extension-theorem" , there is a basis $A'$ of $V$ such that $A \subseteq A'$. Is the reverse true ? that is if $B$ is any basis of $V$ , does ther... | No. Suppose that $V$ is a $k$-dimensional vector space over some infinite field, such as $\mathbb{R}$ or $\mathbb{C}$. Then since every basis contains exactly $k$ members, it follows that any given basis $B$ has exactly $2^k$ subsets, and thus bases for $2^k$ different subspaces of $V$.
But since $V$ has infinitely m... | {
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Solving integral $\int\frac{\sin x}{1+x\cos x}dx$ How I can find the anti-derivative?
$$\int\frac{\sin x}{1+x\cos x}dx$$
| Sorry for “cheating”, but it is as it seems: Wolfram|Alpha states it is not solvable “in terms of standard mathematical functions” (which should then be true).
http://www.wolframalpha.com/input/?i=%E2%88%ABsinx%2F%281%2Bxcosx%29dx
Are you sure this is the correct integral?
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Automorphisms of group extensions Assume we have a group extension $1 \to N \to G \to H \to 1$, and an automorphism $\phi: G \to G$. Is it correct that this automorphism induces automorphisms $\phi_N : N \to N$ and $\phi_H : H \to H$ ?
If so, this would mean that the image by $\phi$ of elements of the form $(n,1_H) \in... | Not true. Take an automorphism which is not inner. For abelian examples, take $G$ to be the points of plane under vector addition, and $N$ to be any line through origin.
Now rotations of the plane by any angle (not $0$ or $\pi$) is an automorphism which does not take $N$ to $N$.
| {
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"timestamp": "2023-03-29T00:00:00",
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Laplacian $\frac{1}{r^2}\frac{\partial}{\partial r}(r^2 \frac{\partial \phi }{\partial r})= \frac{1}{r} \frac{\partial ^2 }{\partial r^2}(r \phi )$ Does anyone have any intuition on remembering or very quickly deriving that
$$\frac{1}{r^2}\frac{\partial}{\partial r}(r^2 \frac{\partial \phi }{\partial r}) = \frac{1}{r} ... | Maybe you want to take a look at how the Laplacian is derived for any system of curvilinear coordinates:
http://en.wikipedia.org/wiki/Curvilinear_coordinates
| {
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Find $m$ and $n$ Two finite sets have m and n elements. Thew total number of subsets of the first set is 56 more than the two total number of subsets of the second set. Find the value of $m$ and $n$.
The equation to this question will be $2 ^ m$ - $2 ^ n = 56$.
But I don't know how to solve this equation.
| Let $k = m-3$ and $l=n-3$, then
$$
2^k-2^l = 56/2^3 = 7.
$$
Now determine the values of $k$ and $l$.
| {
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Using Polar Integrals to find Volume of surface Here's the Question and the work that I've done so far to solve it:
Use polar coordinates to find the volume of the given solid.
Enclosed by the hyperboloid $ −x^2 − y^2 + z^2 = 61$ and the plane $z = 8$
Ah MathJaX is confusing by the way. Not sure how to close the text w... |
Notice that $-x^2-y^2+z^2=61$ can be rewritten as $x^2+y^2=z^2-61$ or
$$x^2+y^2=\sqrt{z^2-61}^2$$
Given a horizontal slice of the upper sheet of the hyperboloid (i.e. given $z$), it should be plain to see from the graph (and the equation that we have a circle of radius $\sqrt{z^2-61}$.
The vertex of the upper sheet oc... | {
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Lie algebra of $\mathbb{R}^{n}$ Until now the only example of lie groups I have seen are subgroups of $GL_n$. Today I had the idea, that also $G=(\mathbb R^n,+)$ must be a lie group ($(\mathbb R^n,+)$ is a group with the differentiable group operation $+$). Is it right that the lie algebra of this lie group is $\mathfr... | Solution 1
The left translation maps have the form $L_a:\mathbb R^n \rightarrow \mathbb R^n: x\mapsto a+x$. So $D L_a(x)=\operatorname{id}$ for all $a,x\in\mathbb R^n$ ($D L_a$ is the total derivative of $L_a$). So the set of all left-invariant vector fields is the set of all constant vector fields $\mathfrak g=\{f:\ma... | {
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"answer_id": 0
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Error solving "stars and bars" type problem I have what I thought is a fairly simple problem: Count non-negative integer solutions to the equation
$$x_1 + x_2 + x_3 + x_4 + x_5 = 23$$
such that $0 \leq x_1 \leq 9$.
Not too hard, right? Simply ignore the upper-bound, count the
$$\begin{pmatrix}23 + (5-1) \\ (5-1)\end{pm... | Your method is correct, there are 15170 solutions. It seems like your python script is doing something wrong at the end.
I wrote a C++ program myself to confirm this:
#include <stdio.h>
int main()
{
int a,b,c,d,e,sum=0;
for (a=0; a<10; a++)
for (b=0; b<24; b++)
for (c=0; c<24; c++)
for (d=0; d<24; d++)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/756462",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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Find Normalizing constant let $f(x,\theta)=C_\theta \exp(-\sqrt{x}/\theta)$ where $x$ and $\theta$ are both positive.
Find the normalising constant $C_\theta$. I get $C_\theta=\sqrt{2}/\theta$ but my book says $C_\theta=1/2\theta^2$. Who is right?
| \begin{align}
u & = \sqrt{x}/\theta \\
u^2 & = x/\theta^2 \\
2u\,du & = dx/\theta^2
\end{align}
$$
\int_0^\infty e^{-\sqrt{x}/\theta} \, dx = \theta^2\int_0^\infty e^{-u} \Big( 2u\,du \Big) = 2\theta^2.
$$
The integral can be done by parts, thus:
$$
\int u \Big(e^{-u}\,du\Big) = \int u\,dv=uv-\int v\,du = -ue^{-u}-\int... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/756560",
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"source": "stackexchange",
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Sigma Algebra Measurable R.V I am trying to figure out what random variables are measurable with respect to sigma algebra given by $[1-4^{-n}, 1]$ where $n= 0, 1, 2, ....$ if $[0,1]$ is the sample space. I believe I can do with with indicator functions but I'm not sure how to write this.
Thanks!
| Well, any indicator function won't do. You can see that there is no way to write $(0,25,0.5)$ as an element of the $\sigma$-algebra generated by those sets, and so the indicator of that set won't be measurable. But the set of simple functions which are measurable in that space is most likely dense with respect to any m... | {
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Least Squares of Symmetric Positive Semidefinite Matrices What's the best (in terms of computation time and numerical robustness) way to find the least squares solution of $$Ax = b$$ if $A$ is symmetric and positive semi-definite?
If $A$ were symmetric and positive definite, the Cholesky decomposition would seem to be ... | According to MATLAB's mldivide():
If the Cholesky Decomposition doesn't work you should use LDL Decomposition.
| {
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Create a Huge Problem I am wondering if any problems have been designed that test a wide range of mathematical skills. For example, I remember doing the integral $$\int \sqrt{\tan x}\;\mathrm{d}x$$ and being impressed at how many techniques (substitution, trig, partial fractions etc.) I had to use to solve it successfu... | In terms of an integral question, I tried to come up with a double integral for my students that would test many first year integral techniques. It requires both integration by parts and multiple subsitutions, as well as an understanding of double integral regions. Here it is -
Sketch the region below $y=\sqrt{\sin x}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "32",
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Explanation of recursive function Given is a function $f(n)$ with:
$f(0) = 0$
$f(1) = 1$
$f(n) = 3f(n-1) + 2f(n-2)$ $\forall n≥2$
I was wondering if there's also a non-recursive way to describe the same function.
WolframAlpha tells me there is one:
$$g(n) = \frac{(\frac{1}{2}(3 + \sqrt{17}))^n - (\frac{1}{2}(3 - \sqrt{... | This is called a linear recurrence. Solving them is fairly straightforward, and is explained here: http://en.wikipedia.org/wiki/Linear_recurrence#Solving.
The key thing to note is that if $f_1$ and $f_2$ are solutions of this recurrence, then $f_1 + f_2$ is as well (except for the initial conditions).
The trick is to a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/756931",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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What's the relation between prime spectrum and affine space?
Let $A$ be a ring ,$X$ be the set of all prime ideal of $A$.For each subset $E$ of $A$,let $V(E)$ denoted the set of all prime ideals of $A$ which contain $E$.
we have:
*
*$V(0)=X,V(1)=\emptyset$
*$V(\bigcap_{i \in I} E_i)=\bigcup_{i \in I} V(E_i)$
*$... | This is really only a partial answer that was too long for a comment but I hope it is helpful.
If you consider the prime spectrum of the ring of regular functions on an affine variety $X$, where by affine we mean the zeros of a collection of polynomials over an algebraically closed field, you almost get back the variet... | {
"language": "en",
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Prove that $\{(a,b):a,b\in\mathbb N, a\geq b\}$ is denumerable. If $S=\{(a,b):a,b\in\mathbb N, a\geq b\}$, how do I prove that $S$ is denumerable?
Work: Since $S \subseteq\mathbb{N\times N}$ I know that $S$ is denumerable. But I don't know how to structure the proof clearly. I know that the two theorems : every infinit... | If you're allowed to use than any subset of a denumerable set is denumerable, then just use the fact that $\mathbb{N} \times \mathbb{N}$ is denumerable, and prove it is if necessary by using an injection $f(m,n) = 2^m 3^n$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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prove that if A is a subset of B, B is a subset of C, and C is a subset of A, then A=B and B=C To prove A=B, I must prove that A is a subset of B and B is a subset of A. A is a subset of B is already given. So all that is left is to prove B is a subset of A.
Is it suffice to say that since A is a subset of B, B is a s... | In fact, the problem deals wit two separate issues!
Check transitivity:
$$A\subseteq B\subseteq C\implies A\subseteq C$$
Check antisymmetry:
$$A\subseteq C\subseteq A\implies A=C$$
(These are rather different!)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/757214",
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"source": "stackexchange",
"question_score": "1",
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Markov Chain Ergodic Theorem (Proof references) Where can I find a proof of the erogidc theorem for Markov chains that doesn't use Birkhoff?
The theorem states the following :
Let $(X_n)_{n\in \mathbb{N}}$ be an irreducible and positively recurrent Markov chain in a countable state space $E$ with invariant measure $\p... | You can find an elementary proof in Durrett's Essentials of Stochastic Processes, at the end of Chapter 1.
| {
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"timestamp": "2023-03-29T00:00:00",
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Finding power series of function could anyone help me answer question?
$$F(x)=\ln\left(\dfrac{7+x}{7-x}\right)$$ Find a power series representation for the function.
| When I was young (that is to say a loooong time ago), one of the series the professor asked us to always remember (because of its extreme simplicity) is precisely $$\ln\left(\frac{1+x}{1-x} \right)=2\sum_{n=0}^{\infty}\frac {x^{2n+1}}{2n+1}$$ Thank you for remembering me my youth
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/757376",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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The splitting fields of two irreducible polynomials over $Z / p Z$ both of degree 2 are isomorphic $p$ is a prime. Let $ f_1, f_2 \in Z / p Z [t]$ both of degree 2 and irreducible. Show that they have isomorphic splitting fields.
My approach was let $ K_1 = F(\alpha_1, \beta_1) / F$ be the splitting field of $f_1$, an... | Some ideas (hopefully you've already studied this stuff's details):
For a prime $\;p\;$ and for any $\;n\in\Bbb N\;$, prove that the set of all the roots of the polynomial $\;f(x):=x^{p^n}-x\in\Bbb F_p[x]\;$ in (the, some) algebraic closure $\;\overline{\Bbb F_p}\;$ of $\;\Bbb F_p:=\Bbb Z/p\Bbb Z\;$ , with the usual op... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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What are good resources for learning Numerical methods for Partial Differential Equations? I'm having an undergraduate course on Numerical Solutions to Ordinary and Partial Differential Equations. I need online resources to supplement my study preferably videos and books. I want to build a good understanding of the sub... | Try:
*
*Numerical Solution of Partial Differential Equations: Finite Difference Methods , by G. D. Smith
Also, use the open courseware at:
*
*MIT Open Courseware
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Properties of compact set: non-empty intersection of any system of closed subsets with finite intersection property Let $X$ be a Hausdorff topological vector space. Let $C$ be a nonempty compact subset of $X$ and $\{C_\alpha\}_{\alpha \in I}$ be a collection of closed subsets such that $C_\alpha \subset C$ for each $\a... | The more general fact is true. If $(C_\alpha)_{\alpha\in I}$ is a collection of closed subsets with finite intersection property, then all these subsets have a common point.
Assume $\cap_{\alpha\in I}C_\alpha=\varnothing$, then for open subsets of $C$ which we denote $U_\alpha=C\setminus C_\alpha$ we have $\cup_{\alpha... | {
"language": "en",
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Show that $(x + 1)^{(2n + 1)} + x^{(n + 2)}$ can be divided by $x^2 + x + 1$ without remainder I am in my pre-academic year. We recently studied the Remainder sentence (at least that's what I think it translates) which states that any polynomial can be written as $P = Q\cdot L + R$
I am unable to solve the following:
... | Suppose $a$ is a root of $x^2+x+1=0$, then we have both $$a+1=-a^2$$ and $$a^3=1$$
Let $f(x)=(x+1)^{2n+1}+x^{n+2}$ then $$f(a)=(-a^2)^{2n+1}+a^{n+2}=-a^{4n+2}+a^{n+2}=-a^{n+2}+a^{n+2}=0$$
Since the two distinct roots of the quadratic are also roots of $f(x)$ we can use the remainder theorem to conclude that the remaind... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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For which values of the real parameter the following... How should I solve this exercise:
For which values of real parameter $a$ the following equality is true:
$$\lim_{x\to 0}{1-\cos{ax}\over x^2}=\lim_{x\to \pi}{\sin{x}\over \pi-x}$$
| As pointed out by Claude Leibovici. Rewrite
$$
\lim_{x\to \pi}{\sin{x}\over \pi-x}=\lim_{\pi-x\to 0}{\sin{(\pi-x)}\over \pi-x}=1.\tag1
$$
As pointed out by Your Ad Here, you will get
$$
\begin{align}
\lim_{x\to 0}{1-\cos{ax}\over x^2}&\stackrel{\text{l'Hospital}}=\lim_{x\to 0}{a\sin{ax}\over 2x}\\
&\stackrel{\text{l'H... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Convergence of the series $\sum_{n=0}^\infty \frac{1}{n+1}\sin\bigr(\frac{p\pi u_n}{q}\bigl)$
Let $(u_n)_{n\in \mathbb{N}}$ defined by : $u_0=1, u_1=1$ and for all integer $u_{n+1}=3u_n-u_{n-1}$
Study the convergence of $$\displaystyle\sum_{n=0}^\infty \frac{1}{n+1}\sin\left(\frac{p\pi u_n}{q}\right)$$ with $p,q \... | When $u$ is an integer, $\sin(u\pi/q)$ only depends on the value of $u$ modulo $2q$. So in your case, only the value of $u_n$ modulo $2q$ is important.
If you look at the sequence $v_n = u_n \pmod {2q}$, since we still have $v_{n+1} = 3v_n - v_{n-1}$ (as well as $v_{n-1} = 3v_n - v_{n+1}$), and because $v_n$ can only t... | {
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Why does the result follow? How does this theorem follow?
Theorem. If $g$ is differentiable at $a$ and $g(a) \neq 0$, then $\phi = 1/g$ is also differentiable at $a$, and $$\phi'(a) = (1/g)'(a) = -\frac{g'(a)}{[g(a)]^2}.$$
Proof. The result follows from $$\frac{\phi(a+h)-\phi(a)}{h} = \frac{g(a)-g(a+h)}{hg(a)g(a+h)}.$... | The results follow because:
*
*if $\lim\limits_{h\to 0}\dfrac{\phi(a+h)-\phi(a)}{h}=\ell$ exists ($\in\mathbb{R}$) then $\ell=\phi'(a)$.
*Calculate the limit of this ratio $\dfrac{\phi(a+h)-\phi(a)}{h}=\dfrac{g(a)-g(a+h)}{h}\cdot \dfrac{1}{g(a)g(a+h)}$ knowing that $g$ is defirentiable at $a$.
*You find that $\lim... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Critical points and Convexity? Function $f(x)$ has no critical points in $M$, can we say $f(x)$ is either convex or concave over $M$?
| If $f(x) = 2x+\sin(x), x\in \mathbb{R}$, then $f^\prime(x) = 2+\cos(x)$ is nowhere $=0$, hence there is no critical point. However, $f^{\prime\prime}(x)=-\sin(x)$ changes sign, so $f$ is neither convex nor concave. By scaling you can do that on any interval, as small as you like.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to prove $x^3$ is strictly increasing I am trying to use $f(x)=x^3$ as a counterexample to the following statement.
If $f(x)$ is strictly increasing over $[a,b]$ then for any $x\in (a,b), f'(x)>0$.
But how can I show that $f(x)=x^3$ is strictly increasing?
| You want to show that the function $f(x) = x^3$ is strictly increasing on $\mathbb{R}$.
Maybe you can just use the definition. That is, let $a < b$. Assume that $0<a$ (you can do the other cases I am sure). Let $h = b - a < 0$. You want to show that $f(a) < f(b)$.
So
$$\begin{align}
f(a) = a^3 &= (b - h)^3\\ &= b^3 - 3... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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c0mpatible system $A^TAx=A^Tb$ Let $A\in\mathbb{R}^{n\times n}$ be a singular matrix. Prove that the system
$$A^TAx=A^Tb$$
is compatible for any $b\in\mathbb{R}^n$.
I want to prove that $A^Tb\in Ran(A^TA)$,i.e. $A^Tb\bot Ker(A^TA)$
| Equivalent statements:
*
*$x\in\mathrm{ker}(A^TA)$,
*$Ax=0$.
Proof: $x\in\mathrm{ker}(A^TA)$ implies that $x^TA^TAx=(Ax)^T(Ax)=\|Ax\|_2^2=0$. Hence $Ax=0$. The other direction is trivial.
Hence $A^Tb\perp\mathrm{ker}(A^TA)$, because $y^TA^Tb=(Ay)^Tb=0^Tb=0$
for any $y\in\mathrm{ker}(A^TA)$.
| {
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Splitting an Indefinite Matrix into 2 definite matrices I'm attempting to use some quadratic programming techniques to solve a particular optimization problem and my chosen Objective Function is indefinite.
I've found some texts online which regard splitting the objective function into two components of which one is po... | You can just set $A = Q + kI$ where $k$ is larger than all the negative eigenvalues of $Q$, and put $B = kI$.
| {
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Another Divergent Series Question Suppose the series $$\sum{a_n}$$ diverges where $a_n\ge 0$ and the sequence is monotone non-increasing. If exactly one element is chosen from each interval of size $k$ -- i.e., one element from $[a_0,a_1,...,a_{k-1}]$, one element from $[a_k,...,a_{2k-1}]$, etc. -- must this series div... | Clearly there is some choice of elements from each interval for which the sub-series diverges, or else the overall series would converge. Now note that for any choice of element, you can bound that element as being greater than or equal to the element that was chosen from the next interval. Thus, for any choice of elem... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Basic question about dimensionality of Euclidean group I have a basic question about the dimensionality of the Euclidean group.
Why are degrees of freedom greater than the dimension? I thought that a degree of freedom is the same as a dimension, as in, $x\;\text{-}\;y$ plane is of dimension $2$ and therefore has $2$ de... | First of all, each Euclidean transformation (an element of $E(n)$) has the form $Ax+b$ where $A\in O(n)$ (an orthogonal matrix) and $b\in R^n$. Clearly, $b$ depends on $n$ parameteris and we just have to compute the dimension of $O(n)$. Orthogonality of a matrix means that each column is a unit vector and any two disti... | {
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Interesting association between tangent lines of slope one and ellipses Why is it that a tangent line with slope $1$ to an ellipse centered at the origin will have a transformation of $\pm \sqrt{a^2 +b^2}$ where $a$ and $b$ are the major and minor axis of the ellipse?
For example: The tangent line of slope one to the e... | The relation you are looking for can be derived algebraically. Start with some general line $y=mx+c$ for the tangent. Substitute it into the equation of your ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. This gives:
$$
\frac{x^2}{a^2} + \frac{(mx+c)^2}{b^2} = 1 \\
b^2 x^2 + a^2 (m^2 x^2 + 2mxc + c^2) = a^2 b^2 \\
(b... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Using Rolle's Theorem to show that $3^x+4^x=5^x$ iff $x=2$ I need to use Rolle's Theorem to show that the only real solution to $3^x+4^x=5^x$ is $x=2$. Here's what I have:
Proof: Note that a number $x$ satisfies $3^x+4^x=5^x$ if and only if $f(x)=0$ where $f(x)=3^x+4^x-5^x$. Obviously $x=2$ is solution since $f(2)=0$. ... | What about the function $(3/5)^x+(4/5)^x-1$. The derivative of this does not have a root.
| {
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"timestamp": "2023-03-29T00:00:00",
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How exactly does the response "infintely many" answer the question of "how many"? I admit that the level of this question is roughly about middle school, but this is what the question asks:
The ratio of nickels to dimes to quarters is 3:8:1. If all the coins were dimes, the amount of money would be the same. Show that... | Three nickels and a quarter make up $40$ cents, as do four dimes. As a consequence, the second sentence in your problem does not amount to an additional condition. It follows that any multiple of the package "$3$ nickels, $8$ dimes, and $1$ quarter" solves the problem.
| {
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How to prove a set of vectors does not span a space. Ok, so I'm a bit curios as to how you can prove a set does not span a vector space.
For example, let ${S}$ be the vector set
\begin{bmatrix}
1\\
0\\
0\\
0\\
\end{bmatrix}
\begin{bmatrix}
0\\
1\\
0\\
0\\
\end{bmatrix}
\begin{bmatrix}
0\\
0\\
1\\
0\\
\end{bmatrix}
\beg... | Plug the vectors in as rows in a matrix, then row-reduce to find a basis for the row space. Remember the row space of a matrix is the subspace spanned by the initial row vectors. If you end up with one or more rows of zeros after row-reduction, then that indicates that your initial row vectors were not linearly indep... | {
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Is a knot $K$ and it's mirror image $^*K$ considered the same knot in terms of tabulating prime knots? If so, why? I'm just wanting to confirm whether this is the case and why? Is it purely to do with the sheer number of knot projections that would have to be dealt with?
| A knot and its mirror image are not always the same, but when it comes to knot tables, most people do not bother drawing both. I believe this is just for space and the fact that we can pretty much visualize what the mirror image looks like. But there is a way to tell.
Most tables will then provide the Jones Polynomia... | {
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"answer_count": 1,
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} |
Looking for notation of set of all entries of some matrix? I'm busy writing my thesis, and I'm looking for some concise notation to denote the supremum of the matrix entries of, say $A \in M_n(\mathbb{R})$. How should I do this?
Looking for something like
$$\sup_{a_{i,j} \in A}|a_{i,j}|$$
but the notation $a_{i,j} \in... | Although not really a notation but a combination of notations, the quantity in question is $\|\operatorname{vec}(A)\|_\infty$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/759087",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Order of groups and group elements? Let G be a group and let p be a prime. Let g and h be elements of G with order p.
I am wondering how I can use group theory to find the possible orders of the intersection between $\def\subgroup#1{\langle#1\rangle}\subgroup g$ and $\subgroup h$ and also to prove that the number of el... | As noted by Potato, the first thing to notice is that the intersection of two subgroups, $\langle g\rangle $ and $\langle h\rangle$ is a subgroup of $G$, but moreover, it is also a subgroup of both $\langle g\rangle$ and $\langle h\rangle$. A cyclic group of prime order, such as $\langle g\rangle$ only has two subgroup... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/759185",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Can anyone explain a residue in fairly simple terms? I'm studying Complex Analysis and everything up to this point has been pretty straightforward to visualise, but I can't get my head around residues, especially as they seem to have two very different definitions (as a Laurent series coefficient and as an expression i... | To see why they equate, just take the following contour integral around the unit circle:
$$
\oint \frac{dz}{z^k}=\int_{0}^{2\pi}\frac{d(e^{i\theta})}{e^{ik\theta}}=\int_{0}^{2\pi}\frac{ie^{i\theta}d\theta}{e^{ik\theta}}=\int_{0}^{2\pi}ie^{i(1-k)\theta}d\theta.
$$
If $k=1$, then the integral is $2\pi i$. If it's any ot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/759315",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Integral $\int_0^{\pi/2} \theta^2 \log ^4(2\cos \theta) d\theta =\frac{33\pi^7}{4480}+\frac{3\pi}{2}\zeta^2(3)$ $$
I=\int_0^{\pi/2} \theta^2 \log ^4(2\cos \theta) d\theta =\frac{33\pi^7}{4480}+\frac{3\pi}{2}\zeta^2(3).
$$
Note $\zeta(3)$ is given by
$$
\zeta(3)=\sum_{n=1}^\infty \frac{1}{n^3}.
$$
I have a previous pos... | From Table of Integrals, Series, and Products Seventh Edition by I.S. Gradshteyn and I.M. Ryzhik equation $3.631\ (9)$ we have
$$
\int_0^{\Large\frac\pi2}\cos^{n-1}x\cos ax\ dx=\frac{\pi}{2^n n\ \operatorname{B}\left(\frac{n+a+1}{2},\frac{n-a+1}{2}\right)}
$$
Proof
Integrating $(1+z)^p z^q$, for $p,q\ge0$, in the $z=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/759513",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
"answer_count": 1,
"answer_id": 0
} |
Smooth approximation of maximum using softmax? Look at the Wiki page for Softmax function (section "Smooth approximation of maximum"): https://en.wikipedia.org/wiki/Softmax_function
It is saying that the following is a smooth approximation to the softmax:
$$
\mathcal{S}_{\alpha}\left(\left\{x_i\right\}_{i=1}^{n}\right... | This is a smooth approximation of maximum function:
$$
\max\{x_1,\dots, x_n\}
$$
where $\alpha$ controls the "softness" of the maximum. The detailed explanation is available here: http://www.johndcook.com/blog/2010/01/13/soft-maximum/
Softmax is better then maximum, because it is smooth function, while $\max$ is not sm... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/759637",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Book recommendation for Linear algebra. I am looking for suggestions, it has to be a self study book and should be able to relate to applications to real world problems.
If it is more computer science oriented , that would be great.
| There is an innovative course Coding the Matrix offered by Philip Klein which consists of a book and a course offered on Coursera and other places. It even has a Twitter account for keeping updated. The reviews are controversial, see also here and here, but it looks as an interesting challenge to try. It is designed, a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/759717",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 7,
"answer_id": 1
} |
What is a good topic for an essay on applications of Calculus 3? In a class I have, the professor has offered extra credit for 1 page paper on a topic in Calculus 3 that has an application in the real world. I know calculus is used a lot in physics but I do not know physics very well.
What is a good topic that is unde... | I have two ideas for you. Since you are only allowed to write one page, you are not going to be able to do much. But may
*
*Look at electromagnetism and Maxwell's equations. Some completely random notes: http://www.phys.ufl.edu/~thorn/homepage/emlectures1.pdf. Take a look at chapter 2.
*Again, since you just have o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/759788",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Probability, random line up Five distinct families arrive to a party. Each family consists of 3 people. The 15 participants of the party are arranged randomly in a line.
Let X be the number of families that their members sit next to each other. Find E[X] and Var(x).
My attempt: Just go straight to find out the pmf of... | Hint: The random variable takes values $0,1,2, \cdots 5$. Find the probability of each event. Also, $X^2$ takes values $0,1,4,16,25$ with the same probabilities, as computed above, and $Var(X)=E(X^2)-E(X)^2$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/759873",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Distance and speed of two people walking 100 miles This is for GRE math prep. Can you explain why the answer to this is 54? Five hours after Sasha began walking from A to B, a distance of 100 miles, Mario started walking along the same road from B to A. If Sasha's walking rate was 2 miles per hour and Mario's was 3 mil... | Let $t$ be the number of hours that Mario walks before they meet. Then Sasha has walked $5+t$ hours, and
$$2(5+t)+3t=100.$$
Or else, without "algebra": Sasha has walked $10$ miles before Mario sets out, so at that time they are $90$ miles apart. Then distance between them shrinks $5$ miles per hour, so it takes $18$ ho... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/759978",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How find this $5xy\sqrt{(x^2+y^2)^3}$ can write the sum of Four 5-th powers of positive integers.
Find all positive integer $x,y$ such
$$5xy\sqrt{(x^2+y^2)^3}$$ can write
the sum of Four 5-th powers of positive integers.In other words: there exst $a,b,c,d\in N^{+}$ such
$$5xy\sqrt{(x^2+y^2)^3}=a^5+b^5+c^5+d^5$$
Thi... | Something that might help. Using what you said, namely:
$x^2+y^2=k^2$, with $x=m^2-n^2$, $y=2mn$ then the left hand side of the equation becomes:
$LHS=10(m^9n-mn^9+2m^7n^3-2m^3n^7)$
Since $RHS\equiv 0$ modulo 2 and modulo 5 then we have $a+b+c+d\equiv 0 \mod{2}$ and $a+b+c+d\equiv 0 \mod{5}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/760071",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
Continuity of bilinear maps Given a vector space $V$ over $\mathbb{R}$ with a norm $||*|| $. Can $(x,y)\rightarrow(x+y)$ be an example of continous bilinear map, if yes, can you please exlain why?
Definition of continuous bilinear map $\lambda$ on $V\times V \rightarrow V$ is:
For all $v,w\in V$, there is $C>0$ such th... | Your map $\lambda:V\times V\rightarrow V$ is continuous, but not bilinear:
For $\mu\not=0\in \mathbb{R}$ and $v,w\not=0\in V$:
$$\lambda(\mu v,w)=\mu v+ w\not = \mu (v+w)=\mu\cdot\lambda(v,w)$$
However, $\lambda$ is a linear map from the vector space $V\times V$ to $V$. Therefore it is continuous if and only if there e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/760248",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Quadratic equations and inequalities $\sqrt{4n+1}<\sqrt{n} + \sqrt{n+1}<\sqrt{4n+2}$ and $[\sqrt{n}+\sqrt{n+1}] = [\sqrt{4n+1}]$ For every positive integer $n$, prove that
$$\sqrt{4n+1}<\sqrt{n} + \sqrt{n+1}<\sqrt{4n+2}$$
Hence or otherwise, prove that $[\sqrt{n}+\sqrt{n+1}] = [\sqrt{4n+1}]$, where $[x]... | Observe that $$\lfloor\sqrt{4n+1}\rfloor=\lfloor\sqrt{4n+2}\rfloor$$
unless $4n+2$ is perfect square
But any square $\equiv0,1\pmod4$
$$\implies\lfloor\sqrt{4n+1}\rfloor=\lfloor\sqrt n+\sqrt{n+1}\rfloor=\lfloor\sqrt{4n+2}\rfloor$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/760330",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Evaluate $\displaystyle\lim_{j\to0}\lim_{k\to\infty}\frac{k^j}{j!\,e^k}$ I found this problem in my deceased grandpa's note today when I was visiting my grandma's home.
\begin{equation}
\lim_{j\to0}\lim_{k\to\infty}\frac{k^j}{j!\,e^k}
\end{equation}
I asked my brother and he said the answer is $\cfrac{1}{2}$, but as us... | The sum was probably transcribed wrong somewhere along the way,because it doesn't make much sense as written (the limit as $j \rightarrow 0$
of something involving $j!$ ?). But since the answer is supposed to be $1/2$,
I'm guessing that the intended formula was something like
$$
\lim_{k \rightarrow \infty} \sum_{j=0}^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/760451",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Estimate standard deviation of sample You have a random sample of 25 objects with mean weight of 24 grams, estimate the standard deviation of the sample.
In addition, you know it's supposed to be 25 grams with a deviation of 1 gram, but this has no relevance to the above question, right?
How is this done? Looking in my... | A sample mean of $24$ is unlikely from a sample of twenty-five items with a population mean of $25$ and population standard deviation of $1$. So the sample casts doubt on the population parameters.
But conditioned on the data given and ignoring issues such as finite populations, a good estimate of the variance of the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/760526",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Looking for an introductory Algebraic Geometry book I am looking for recommendations on an AG text to work through this summer, possibly with the help of a mentor. I would want this book to have some introduction to categories, and then develop the modern methods (some development of sheaves and schemes), hopefully up ... | These notes by Andreas Gathmann are precisely what you're asking for. He starts very gently, schemes being introduced in Chapter 5, but he ends with sheaf cohomology (including Riemann-Roch) and some intersection theory.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/760579",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
The eigenvalue of $A^TA$ If $\lambda$ is the eigenvalue of matrix $A$,what is the eigenvalue of $A^TA$?I have no clue about it. Can anyone help with that?
| Generally, you won't be able to say much about them.
However, if $A$ is for instance real symmetric, it is diagonalizable with real eigenvalues, meaning there is an orthogonal matrix B (that is with $B^{-1}=B^T$) and a diagonal matrix D such that:
$$A = B D B^T$$
$$A^TA = (B D B^T)^T B D B^T = B D^T B^T B D B^T = BD^2B... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/760654",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
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