Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Find continuos function $x$ with $x(0)=0$ such that $\|x-y\| \geq 1$ where $y(0)=0$ and $\int_0^1 y(t) dt = 0$ Consider $C[0,1]$ space of continuos functions with the uniform norm. Let $X = \{ x \in C[0,1]: x(0)=0\}$ and $Y = \{ y \in X: \int_0^1 y(t) dt = 0 \}$ subspaces of $C[0,1]$. How can I show that $\exists x \i... | Let us suppose there exists such a function $x$. We define $m$ by:
$$m=\int_0^1 x(t) dt$$
the idea is to show that $|m|\geq1$ which is in contradiction with $x$ continuous, $x(0)=0$ and $\|x\|_\infty=1$.
Let $\epsilon >0$. There exist $g_\epsilon \in X$ such that:
$$\int_0^1 g_\epsilon(t) dt=1-\epsilon, \, \|g_\epsilon... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2762206",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
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Properties of inclusion map between topological spaces. Let $X$ be a topological space and $Y$ a subset of $X$. Write $i: Y \to X$ for the inclusion map. Choose the correct statement:
*
*If $i$ is continuous, then $Y$ has the subspace topology.
*If $Y$ is an open subset of $X$, then $i(U)$ is open in $X$ for all su... | I came across a map like: (X, T1) and (X, T2) is a topological space with topologies T1 and T2 where T1 is stronger than T2. Can we say that a map from (X, T1) to (X, T2) is an inclusion map? Anybody please.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2762322",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
} |
Does there always exist a continuous/smooth map from $\mathbb{R}^n$ onto a manifold $M^n$? I want to know if we can get away with only having one set of coordinates for a manifold if we allow multiple coordinates to map to the same point.
| Another construction goes through Riemannian metrics. Assume that $M$ is connected. If you put a complete riemannian metric on your manifold, the exponential map $\exp_p:T_pM\rightarrow M$ will be surjective. The idea is that any two points are connected by a geodesic. You can view the world standing in one point!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2762450",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
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Preservation of closed sets under linear transformation. Let $f:\mathbb{R^n} \rightarrow \mathbb{R^m}$ and let $A$ be a closed set in $\mathbb{R^n}$. I would like to know if $f(A)$ is a closed set.
I know this question is pretty much the same as this one: A linear transform of a closed set is closed. But the answer giv... | For posterity, a necessary and sufficient when $A$ is an arbitrary subset of $\mathbb{R}^n$ and $f: \mathbb{R}^n \to \mathbb{R}^m$ is linear, is:
$$
f(A) \textrm{ is closed} \iff A + \textrm{ker}(f) \textrm{ is closed.}
$$
This is a corollary of the infinite dimensional version which can be found, for example, in Holme... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2762578",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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I'm confused about exactly how a subgroup of $\Bbb Z \times \Bbb Z$ can be isomorphic to $\Bbb Z$ I have read that there exists a subgroup $H$ of $\Bbb Z \times \Bbb Z$ such that $H \cong \Bbb Z$.
If $\Bbb Z \times \Bbb Z:=\{(a,b)|a,b \in \Bbb Z\}$
With the group operation $(a,b)+(c,d)=(a+c, b+d)$.
Then by defining a ... | Think about the "lines" of $\mathbb{Z} \times \mathbb{Z}$, i.e
$$\{(an, bn) \mid n \in \mathbb{Z}\}$$
with $a, b \in \mathbb{N}_0$ fixed.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2762684",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finding all complex entries? Find all complex triples $(x, y, z) $such that the following matrix is diagonalizable
$A = \begin{bmatrix}1&x&y \\ 0&2 & z \\0&0&1\end{bmatrix}$
my attempts :
matrix A is an upper triangular matrix.
so the the eigenvalues of A are diagonal entries $1,2,1$
This implies that $A$ is diagonal... | This is equivalent to the fact that
$$
A-I_3 =
\begin{bmatrix}
0 &x&y \\
0&1 & z \\
0&0&0
\end{bmatrix}
$$
is such that $dim(ker(A-I_3))=2$ which is, in turn, equivalent to
$$
dim(Im(A-I_3))=1
$$
and the fact that the two columns
$$
\begin{bmatrix}
x&y \\
1 & z \\
0&0
\end{bmatrix}
$$
are proportional. You fin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2762795",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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How is the area of a parallelogram given by |(c1-c2)(d1-d2)/ (m1-m2)|? My book gives this formula for the area of a parallelogram bounded by the lines $$y = m_1x + c_1,\ \ y = m_1x + c_2 \,\ \ y= m_2x + d_1, \text{ and } \ y = m_2x + d_2$$is given by $$\operatorname{abs}\left(\frac{(c_1-c_2)(d_1-d_2)} {m_1-m_2}\right)... | This i because c1-c2 is not the height. If you want to determine the height between $y = m_1x + c_1$ and $y = m_1x + c_2$ you've to compute the minimal distance between them.The distance between the two lines is the distance between the two intersection points of these lines with the perpendicular line. So the height i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2762923",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Sequence and series with specific nth term Let $a(1) = 2$ , $a(n+1) = a(n)^2-a(n) + 1$ for $n\geq 1$, Find $$\sum_{n=1}^{\infty} \frac{1}{a(n)}$$
| (not an answer)
I calculated with Python:
def f(x): return x**2 -x +1
def list_sum(a):
l = []
term = a
for k in range(20):
l.append(1.0/term)
term = min(f(term),50000000000)
print "term=", term
return sum(l)
print list_sum(2)
and the answer seems to be quite close to $1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2763008",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proving The Limit Is Unique Using Open Balls
Prove: The Limit Is Unique
Attempt, let assume that the sequence $\{x_m\}$ has two limit namely: $p ,q$
Let $\epsilon=\frac{d(p,q)}{2}$
$p,q$ are limits of $\{x_m\}\Rightarrow$ There are $m_1\text{ and }m_2$ such that for all $n_1>m_1$ and $n_2>m_2$ let $N=\max\{n_1,n_2\}... | No, because you've made a typo in "we get $x_N \in B(p,\epsilon)$ and $x_N \in B(\color{red}{p},\epsilon)$", so in your proof, there's no relation between $N$ and $q$. To fix this, change $\color{red}{p}$ to $q$ so that the strict inequality following $(1)$ is valid.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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in general, assuming that the solution has a Laplace transform, aren't we restricting ourselves to a subset of the solution set of the ODE? My professor in a lecture on ODEs, stated the following:
Consider the following ODE $$y'' - 2y = f(x),$$ where the Laplace
transform of $f(x)$ exists.
To solve this ODE, we can ... | A priori yes, we could be missing solutions this way, but it's not hard to prove that it's actually ok.
Say $E$ is the space of functions defined on $(0,\infty)$ that "have Laplace transforms", meaning that there exist $c$ and $a$ so $|f(t)|\le ce^{at}$. For the specific DE you mention we need to show that if $y''-2y\i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2763218",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that a polynomial ring over R in infinitely many variables is non-Noetherian. I am struggling with understanding this proof. I am confused as to why they took $a_{n+1}$ and how they got $a_{n+1}= b_1a_1+...+ b_na_n$, $a_1 = ... = a_0 = 0,$ and $a_{n+1} = 1$ gives $1 = 0$ in $\mathbb{R}$. Please help with explainin... | If $S=\mathbb{R}[a_0,a_1,\dots]$ were Noetherian then any chain of ideals $$I_0\subset I_1\subset\dots$$ would stabilize, that is, for some $n$ we would have $$I_n=I_{n+1}=I_{n+2}=\dots$$ To prove that $S$ is not Noetherian, we first assume that it is, and then see if we get a contradiction.
First we need a chain of i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2763317",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Area Between Two Circles - Analytic Geometry I've been wondering if the following has a generalisation, and if so, what is it? I couldn't arrive at a result, partly because the calculation was too much to do and I didn't know in what direction to proceed.
Consider two circles,
$ (x-a)² + (y-b)² = c² $ and $ (x-p)² + ... | First, find points of intersection, which will be the solutions of this system: $$\begin{cases}
(x-a)^2+(y-b)^2=c^2 \\
-2ax-2by+2px+2qy=c^2-r^2-a^2-b^2+p^2+q^2 \\
\end{cases}
$$
Once you found them, find the length of chord that connects intersection points. Let's say the chord length is $l$. The common area will be tw... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2763390",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If $\beta$ is the transposition $\beta = (1, 4)$, compute both $\beta\alpha$ and $\beta\alpha\beta^{−1}$ and compute orders. Let $\alpha$ be the permutation in the symmetric group $S_9$ defined by
$$\alpha =\left(\begin{matrix}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9\\ 5 & 1 & 7 & 6 & 3 & 9 & 2 & 8 & 4\end{matrix}\right)$$
I... | In the symmetric group $S_n$, the group operation is the composition of permutations. The permutation $\beta\alpha$ is the permutation $\alpha$ followed by $\beta$. For example, $(\beta\alpha)(1)=\beta(\alpha(1))=\beta(5)=5$.
To answer your second question, $1,2,...n$ are not elements of $S_n$. The elements are $\texti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2763597",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
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Intuition behind Lagrangian multiplier in this problem What shape should a rectangular box with a specific volume be in order to minimize its surface area?
Let the lengths of the box's edges be x, y, and z. Then the constraint of constant volume is simply g(x,y,z) = xyz - V = 0, and the function to minimize is f(x,y,z)... | The correct problem handling is as follows
First the Lagrangian
$$
L(x,y,\lambda) = 2(x y+x z+y z)+\lambda(V-x y z)
$$
Second the stationary points determination by solving
$$
\nabla L = \{L_x,L_y,L_z,L_{\lambda}\} = 0
$$
or
$$
-\lambda y z + 2 (y + z) = 0\\
-\lambda x z + 2 (x + z) = 0\\
-\lambda x y + 2 (x + y) = 0\\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2763703",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why is this logical formula invalid for expressing the statement "anything bought it not human"? In my first order logic notes, one example that comes up is to express that "anything bought is not human". Is says this can be expressed as:
$\forall x(\exists y \, bought(y,x) \rightarrow \neg human(x))$
where $bought(y,x... | Since we don't care about the real world meaning, but the underpinning logic, we'll simply use the relation $B$ and univariate $C$ to shorten the math statements.
The first statement is of the form: $\forall x~((\exists y~B(x,y))\to\neg C(x))$. Anything will not satisfy $C$ if it is $B$-related to something.
The se... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2763850",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is there a function that is Riemann integrable but not monotonic and not piecewise continuous? I know that continuous functions are Riemann integrable, piecewise functions are Riemann integrable and that monotonic functions are Riemann integrable.
I would like to know if there is a function $f$ that is Riemann integrab... | Yes. Such a function exists. For example, let $f:[0,1]\rightarrow\mathbb{R}$
be defined by $f(x)=0$ if $x\in\{0,1\}\cup\left([0,1]\setminus\mathbb{Q}\right)$
and $f(x)=\frac{1}{q}$ if $x\in(0,1)\cap\mathbb{Q}$ and $x=\frac{p}{q}$,
where $p,q\in\mathbb{N}$ and $p,q$ are relatively prime. $f$ is
known as the Riemann func... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2763944",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Does the existence of a Kuratowski-infinite set imply the existence of a Dedekind-infinite set? A set $X$ is called Dedekind-infinite if there is a injective, non-surjective map $X\to X$ and Kuratowski-infinite if $\mathcal P(X)$ is not generated by $\{\emptyset\}\cup\{\{x\}|x\in X\}$ as a sub-semilattice with respect ... | Yes. Easily, a set which is not Kuratowski finite is not finite. Ant the second power set of an infinite set is Dedekind infinite.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2764037",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Does every distribution with a Lebesgue integrable density also admit a Riemann integrable density? Suppose we have a continuous distribution function $F$ with density $f$ such that $f$ is Lebesgue integrable. Does $F$ then also have a density that is Riemann integrable?
Here's an attempt to construct a counterexample.... | Consider a set $C$ like the standard Cantor set, but of measure $>0$. Take the characteristic function of it. Note that $C$ closed and the interior of $C$ is empty. Moreover, $C$ is such that for every open interval $I$ that intersects $C$ we have $\mu(C\cap I)>0$. Therefore, for every $x \in C$ and $I$ open interva... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2764149",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Finding minimum possible value of $n+z^3$
Let $z$ and $z+1$ be complex numbers such that $z$ and $z+1$ are both $n^{\text{th}}$ complex roots of $1$. If $n$ is a multiple of $5$, compute the minimum value of $n+z^3$.
What I started out with was $(\operatorname{cis}\theta)^n=(\operatorname{cis}\theta+1)^n=1$.
Simplify... | A different way to find the candidates for $z$ and $z+1$ is to note that they are a distance apart of one on a horizontal line yet they both have to lay on the unit circle. It just so happens that fitting an equilateral triangle with one vertex in the center of the circle at the other two on the circle fits the bill.
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2764287",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Are there always harmonic forms locally? Let $M$ be a smooth $d$-dimensional oriented Riemannian manifold, and let $p \in M$.
Let $0 \le k \le d$ be fixed.
Does there exist an open neighbourhood $U$ of $p$, which admit a non-zero harmonic $k$-form? i.e $\omega \in \Omega^k(U)$ satisfying $d\omega=\delta \omega=0$?
S... | Yes, locally there is always an infinite-dimensional space of harmonic $k$-forms (as long as the dimension $d$ is at least 2). One way to see this is a variational approach, since harmonic forms are critical points of an energy functional.
Assume that $U \subset M$ is a small ball around $p$ and that $\alpha$ is some s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2764421",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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How to solve this complex limits at infinity with trig? Please consider this limit question
$$\lim_{x\rightarrow\infty}\frac{a\sin\left(\frac{a(x+1)}{2x}\right)\cdot \sin\left(\frac{a}{2}\right)}{x\cdot \sin\left(\frac{a}{2x}\right)}$$
How should I solve this? I have no idea where to start please help.
| HINT
Note that
$$\frac{a\sin\left(\frac{a(x+1)}{2x}\right)\cdot \sin\left(\frac{a}{2}\right)}{x\cdot \sin\left(\frac{a}{2x}\right)}=\frac{a\sin\left(\frac{a(x+1)}{2x}\right)\cdot \sin\left(\frac{a}{2}\right)}{\frac{a}2\cdot \frac{\sin\left(\frac{a}{2x}\right)}{\frac{a}{2x}}}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2764499",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 0
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Rationality of a power with irrational exponent The following fact is known:
If $a$ and $b$ are both irrational numbers, then $a^b$ can be a rational number.
Proof. Suppose $a^b$ is always irrational. Then $\sqrt{2}^\sqrt{2}$ is an irrational number, which in turn implies that $\left(\sqrt{2}^\sqrt{2}\right)^\sqrt{2}$... | $2^b = 3$ where $b = \log_2(3)$ is irrational. In fact, if $2^b = c$ where $b$ and $c$ are rational, $b$ must be an integer.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2764570",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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Convergence of a series of exponentials Let $\{u_k\}_{k \geq 1}$ be a sequence of positive real numbers such that $u_k \to +\infty$.
For $t>0$, consider the series
$$\sum_{k=1}^{+\infty} e^{-u_k t}.$$
I am wondering if there always exists a $t>0$ such that this series converges (pointwise) ?
| No I don't think so. For $k$ large enough set $u_{k} = \ln (ln k)$. Then
\begin{equation*}
\sum_{k = N}^{\infty}e^{-t(\ln{(\ln k})} = \sum_{k = N}^{\infty} \frac{1}{(\ln{k})^{t}}
\end{equation*}
which is divergent for all $t > 0$ by the Cauchy condensation test.
| {
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"timestamp": "2023-03-29T00:00:00",
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What is the smallest $n = pq$ for which the RSA encryption and decryption works? The example at https://www.di-mgt.com.au/rsa_alg.html#simpleexample claims that the smallest value for the modulus $ n $ for which the RSA algorithm works is $ n = 11 \times 3 = 33 $.
But slide number 30 (page 30) of http://haifux.org/lect... | I've looked at the two papers. Clearly, strength is not the point, even $p,q$ of the order of about a hundred are open to attacks by hand/calculator.
Firstly $e\neq d$ is an absolute requirement for the definition of RSA since in practice $e$ is public and $d$ is private (needs to be kept secret) so both cannot be the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2764796",
"timestamp": "2023-03-29T00:00:00",
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Determine an equation of the line $g$ in coordinates $x', y'$ in terms of the basis $B$
Given is the equation $y = (2+\sqrt{3}) \cdot x \,\,$ which describes
the line $g$. Also, the basis$B = \left\{\vec{b_1}; \vec{b_2}\right\} =
\left\{ \begin{pmatrix} \frac{\sqrt{3}}{2}\\ \frac{1}{2}
\end{pmatrix}; \begin{pmatrix... | The straightforward way to approach this is to express $x$ and $y$ in terms of $x'$ and $y'$ and substitute into the equation. Recalling the definition of coordinates relative to a basis, you have $$\begin{pmatrix}x\\y\end{pmatrix} = x'\begin{pmatrix}{\sqrt3\over2}\\\frac12\end{pmatrix}+y'\begin{pmatrix}-\frac12\\{\sqr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2764950",
"timestamp": "2023-03-29T00:00:00",
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Prove that following recursively defined sequence converges and its limit is 1/2
\begin{cases}a_1 = 1\\ \\ \displaystyle a_{1+n} = \sqrt{\sum_{i=1}^n a_i}\end{cases}
Prove that $\{\frac{a_n}{n}\}$ is convergent and its limit is $\frac12$
My proof: We can recursively define relation between $a_{1+n}$ & $a_n$ as
$a_{... | As pointed out in question, the sequence satisfy the recurrence relation $a_{n+1} = \sqrt{a_n^2 + a_n}$.
From this, it is easy to see $a_n$ is strictly increasing and positive.
One consequence of this is $a_n$ cannot be bounded.
Assume the contrary, then $a_n$ converges to some finite value $M > 0$. Since the map $x \m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2765035",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find the equation of the plane tangent to the surface at a given point I am given the equation of a surface:
$$x^3+y^3+z^3-3xyz =0$$
And I need to find the equation of the plane tangent to this surface at $(1,1,1).$
At first, this task did not look easy for me as we are not given an explicit equation of a surface, but... | The so called surface $f(x,y,z) = x^3+y^3+z^3-3 x y z= 0$ is the product of a plane and a null radius cylinder or
$$
f(x,y,z) = (x+y+z)(x^2+y^2+z^2-x y- y z - x z) = 0
$$
The plane doesn't contain the point $(1,1,1)$ and the cylinder contains it but it has null radius so it is reduced to a line (generatrix) and a line ... | {
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"answer_id": 1
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Why does using "long division" to simplify $1/P(D)$ work when solving $P(D)=x^a$? If you have the differential equation $P(D)=x^a$ you can write it as $(1/P(D))(x^a)$ and use "long division", treating $D^a$ like a regular variable $x^a$ to simplify the operator $1/P(D)$ until the $D^a$ term(because further terms are ze... | This works because the differentiation operator $D$ is in fact an object that can be "added" and "multiplied" just like real numbers can. The same is true of any Linear operator on a vector space. Note that $P$ is a polynomial with constant coefficients. It doesn't work if the coefficients of $P$ are allowed to be func... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2765194",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Does $e^A=e^B$ imply $[A,B]=0$? Let $A$ and $B$ be two matrices, and suppose that $\exp(A)=\exp(B)$. Does this imply that $[A, B]=0$?
BCH's formula shows a clear relation between $\exp(A)\exp(-B)$ and $[A, B]$, but the implication is far from obvious.
However, I haven't been able to find a counterexample for this fact.... | You can cook up lots of matrices with $\exp(A)=I$. Just take $A=2\pi A_0$
where $A_0^2=-I$. For instance $A_0=\pmatrix{0&1\\-1&0}$. Let $B=2\pi B_0$
where say, $B_0=\pmatrix{1&1\\-2&-1}$. Then $B_0^2=-I$ so $\exp(B)=I$
with $B=2\pi B_0$. But $AB\ne BA$ here.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2765287",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Equality between integrable functions on a probability space and improper Riemann integral I have the following proof to do:
Let $X:(\Omega,\mathcal{A},P)\to [0,\infty]$ be integrable. Prove the following
$$\int X\,dP = \int_{0}^{\infty} P(\{\omega:X(\omega)\geq x\})\,dx$$
where the RHS is an improper Riemann integral.... | Hint: write
$$
X(\omega)=\int_0^\infty {\bf1}(x\le X(\omega))\,dx
$$
Substitute this into $\int X(\omega)\,dP$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2765409",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Calculating a determinant. $D_n$=\begin{vmatrix}
a & 0 & 0 & \cdots &0&0& n-1 \\
0 & a & 0 & \cdots &0&0& n-2\\
0 & 0 & a & \ddots &0&0& n-3 \\
\vdots & \vdots & \ddots & \ddots & \ddots&\vdots&\vdots \\
\vdots & \vdots & \vdots & \ddots & \ddots& 0&2 \\
0 & \cdots & \cdots & \cdots &0&a&1 \\
n-1 & n-2 & n-3 & \cdots... | You can just go for calculating the characteristic polynomial $\chi_{-A}$ of minus the matrix $A$ (the one at $a=0$), then your determinant will be $\chi_{-A}[a]$. As you already found that the rank of $A$ is$~2$ (if $n\geq2$; otherwise it is $0$) the coefficients of $\chi_{-A}$ in all degrees less than $n-2$ are zero ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2765475",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 2
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Question about irreducible polynomials over a finite field. If a polynomial $f(x)$ is irreducible over a finite field, does that mean the only factors are $\{1, f(x)\}$?
How would I go about proving a polynomial $f(x)$ is irreducible over a finite field? A bit of searching on StackExchange showed me this:
Irreducibili... | By definition a non-zero non-unit element $r$ in a ring $R$ is irreducible if whenever $d$ divides $r$, $d$ is either a unit, or $d$ is the product of $r$ by a unit. So the set of divisors may be considerably larger and contains all elements of the field.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2765565",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Why do you use the inverse matrix to find the image of a curve in the plane? Given a $2\times2$ linear transformation matrix and the equation of a curve in the plane (e.g. $x^2+y^2=1$), why does one use the inverse matrix to find the equation of the image of the equation after the transformation?
I recently watched a v... | A nice way to think about it is to consider the transformed curve as a geometric locus. Fist of all let the starting curve be $S$ and the transformed curve be $C$ and $\alpha$ the transformation itself. What we need to obtain is a function $f(X,Y)=0$ defined as the set $C$ of points $(X,Y) \implies (x,y) \in S$, namely... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that $\ln{x} \leq \frac{4x}{x+4}$ Can you prove that $\ln{x} \leq \frac{4x}{x + 4}$ for $\forall x > 0$? I can't. I tried using the inequality $\ln{x} \leq x - 1$ as follows:
$$\begin{align} x - 1 & \leq \frac{4x}{x+4} \\
x^2 + 3x - 4 & \leq 4x \\
x^2 - x - 4 & \leq 0 \\
x(x - 1) - 4 & \leq 0
\end{align}$$
so I... | Hint: In the interval $x>0$, the LHS is increasing without bound, while the RHS is increasing with the bound $4$ (the horizontal asymptote $y=4$).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2765745",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving whether $f(x) =|x|^{1/2}$ is uniformly continuous for $f:\mathbb{R}\to\mathbb{R}$ I need to prove following question:
proving whether or not, $f(x) =|x|^{1/2}$ is uniformly continuous for $f:\mathbb{R}\to\mathbb{R}$
My attempt was based on this is not uniformly continuous. following definition and With taking $... | We have the inequality that $\left||x|^{1/2}-|y|^{1/2}\right|\leq|x-y|^{1/2}$ because, say, $|x|\geq|y|$, then $|x|=|y+(x-y)|\leq|y|+|x-y|+2|y|^{1/2}|x-y|^{1/2}=\left(|y|^{1/2}+|x-y|^{1/2}\right)^{2}$, so $\left||x|^{1/2}-|y|^{1/2}\right|=|x|^{1/2}-|y|^{1/2}\leq|x-y|^{1/2}$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Proof Verification: Tao, Analysis I: Exercise 5.5.3. Let $E$ be a subset of $\Bbb{R}$, let $n\geq 1$ be an integer, and let $m,m'\in\Bbb{Z}$ with properties that $\frac{m}{n}$ and $\frac{m'}{n}$ are upper bounds of $E$, but $\frac{m-1}{n}$ and $\frac{m'-1}{n}$ are not upper bounds. Then $m=m'$.
Proof:(by contradiction)... | Your proof seems to be o.k. But the following arguments are simpler:
Suppose that $m'<m$. Then $m-m'>0$. Since $m-m' \in \mathbb Z$, we have $m-m' \ge 1.$ Thus $m' \le m-1$ and therefore $\frac{m'}{n} \le \frac{m-1}{n}.$
Conclusion: $\frac{m-1}{n}$ is an upper bound of $E$. Contradiction !
Hence we have $m' \ge m$. Th... | {
"language": "en",
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Permutations without repeats Consider this matrix:
A B C D E
D C A E B
B A E C D
E D B A C
C E D B A
The letters are arranged so that no row and no coulumn contains the same letter twice (Sudoku style).
Let's call the number of diffent letters $n$ (5 in the above example). When writing the first row, I have $n!$ permu... | When entering the second row, it has to be a derangement of the first row. When $n=5$ there are $44$ derangements. But from the third row onwards
the number of possibilities will depend on exactly what the previous rows
contain.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2766069",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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every open set can be expressed as a countable union of compact sets I'm studying Sard's theorem and I want to know why is true that every open set can be expressed as a countable union of compact sets.
Thank you!
| @harmonicuser has a nice explicit construction for $\mathbb{R}^n$. But as you use the tag general topology:
In fact the statement holds in all hereditarily Lindelöf locally compact Hausdorff spaces $X$ (of which the Euclidean spaces are a prominent example): if $O\subseteq X$ is open, by local compactness plus Hausdorf... | {
"language": "en",
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Are vectors in the null space of a matrix considered eigenvectors? From what I've learned about the definition of an eigenvector, it seems like a vector that gets mapped to zero should just be considered an eigenvector where $\lambda = 0$. Is that true, or are those considered a special case?
| Yes it is correct by the definition, for $\vec x\neq 0$
$$A\vec x=0\vec x$$
then $\vec x$ is an eigenvector with eigenvalue $\lambda=0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2766478",
"timestamp": "2023-03-29T00:00:00",
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Show that there cannot be an entire function that satisfy $|z+(\cos (z)-1)f(z)|\leq 7$ Show that there cannot be an entire function $f(z)$ that satisfy $|z+(\cos (z)-1)f(z)|\leq 7$.
I thought about showing somehow that $f(z)$ is bounded and by Liouville's theorem it is constant, then the inequality does not hold for al... | Suppose $f$ is entire. Then $g(z) = z + ( \cos z - 1 ) f(z)$ is entire also. By Liousville's theorem, the given inequality $|g(z)| \leq 7$ implies that $g$ must be a constant, say $g(z) = c$. But $g(z) = c$ implies that $f(z) = \frac{c - z}{\cos z - 1}$, and this is a problem, for $\cos z - 1 = 0 \iff z \in \{ 2\pi n \... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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convergence in operator norm Let $H$ be Hilbert space, $\{a_n\}_{n=1},$ be ONS and $K$ be a compact operator. Suppose $K_nx:=(Kx,a_n)a_n$, $\sum _{n=1}^{N} K_n$ convergent as $N \to \infty$.
So,$||K_nx||=|(Kx,a_n)|=|(x,K^* a_n)| \leq ||x|| ||K^* a_n||$ and so $\lim ||K_n||\leq \lim ||K^* a_n||=0$ since $K^*$ is compact... | Let $L : H\to H$ be given by
$$ L(x) = \sum_{n=1}^\infty \langle x, a_n\rangle b_n.$$
Then
$$K_N :=\sum_{n=1}^N K_n =L_N \circ K,$$
where $L_n (y)= \sum_{n=1}^N \langle y, a_n \rangle b_n.$ To show that $K_N$ converges, it suffices to show that $L^{-1}\circ K_N$ converges to $K$.But
$$L^{-1} \circ K_N (x)= \sum_{n=... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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On multivariable functions and graphs The definition of graph from Wikipedia is the following:
In mathematics, the graph of a function f is, formally, the set of all ordered pairs $(x, f(x))$, and, in practice, the graphical representation of this set. If the function input $x$ is a real number, the graph is a two-dim... | *
*I have not checked, but those might be the level surfaces of the function of $3$ variables. In principle, any function $$f:\mathbb R^n\to \mathbb R$$ can eventually be reduced to a collection of level curves, namely curves in $\mathbb R^2$.
*Functions $$f':\mathbb R^n\to \mathbb R^m$$ are usually thought of as euc... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Simple proof that if $A^n=I$ then $\mathrm{tr}(A^{-1})=\overline{\mathrm{tr}(A)}$ Let $A$ be a linear map from a finite dimensional complex vector space to itself. If $A$ has finite order then the trace of its inverse is the conjugate of its trace.
I know two proofs of this fact, but they both require linear algebra fa... | If the trouble is just a simple proof for the fact that
$$
\text{tr}\,A=\sum_{i=1}^n\lambda_i
$$
you can try the following approach instead of JNF.
*
*In the field $\Bbb C$ we can factorize $\det(\lambda I-A)=\prod_{i=1}^n(\lambda-\lambda_i)$ where $\lambda_i$ are all eigenvalues (possibly repeated with multiplicit... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Probability of drawing a king immediately after an ace among $5$ cards drawn Five cards are drawn one by one from a standard deck of $52$ cards. What is the probability of drawing a king immediately after an ace?
The number of ways for taking $5$ cards, one by one, from a deck of $52$ is $52.51.50.49.48$
The number of ... | The number of sequences of five cards is
$$P(52, 5) = 52 \cdot 51 \cdot 50 \cdot 49 \cdot 48$$
If an ace immediately precedes a king, there are four positions in the ace-king subsequence can begin, four possible suits for the ace, four possible suits for the king, and $50 \cdot 49 \cdot 48$ ways of selecting the remai... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Making a regular tetrahedron out of concrete I'm trying to make the following tetrahedron made of concrete just for fun:
Each edge is a beam with a triangular cross section.
I imagine the easiest way is to make 6 identical truncated triangular prisms and glue them. Identical because I would need to make only one mold.... | I think you are after the dihedral angle of the regular tetrahedron
which is $\cos^{-1}(1/3)$ or about 70.53 degrees.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2767201",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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"answer_id": 1
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Combinations including "at most" My wife and I cannot figure out how to do a probability question including an "at most" clause. We are given 18 items, 10 of a and 8 of b. If we pick three at random, we need to know how many possibilities of three have at most 2 of b.
We tried finding the probability of having exactly ... | If you can have at most 2 b's, that means the only case that wouldn't count is if all 3 were b's. So, if you can find that probability, you can subtract that from 1 to get the probability of not all 3 being b's, which is equivalent to at most 2 of them being b's.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is the $f(x,y,z)$ is continuous and differentiable at $(0,0,0)$?
Let
$$f(x,y,z)=\begin{cases}
\displaystyle\frac{xyz}{x^2+y^2+z^2}& \text{if $(x,y,z)\neq(0,0,0)$,}\\
0& \text{if $(x,y,z)=(0,0,0)$.}\end{cases}$$
Is $f(x,y,z)$ continuous and differentiable at $(0,0,0)$?
In case of two variables I know that we can ... | Firstly we might check whether or not f is continuos in (0,0,0) by calculating the limit as $(x,y,z)\to(0,0,0)$. If it is discontinuos then it can't be differentiable indeed continuity is a necessary condition since differentiability implies continuity.
And in this case for $(x,y,z)\to(0,0,0)$ by spherical coordinates... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to prove $\frac{1}{n+1}$ is a Cauchy sequence I'm a little stumped by how I should go about proving that $\frac{1}{n+1}$ is a Cauchy sequence. I know that $\frac{1}{n}$ is a Cauchy sequence and understand the proof for that, but don't quite get how I can use that to show this is a Cauchy sequence.
I know that the d... | Hint
$$|a_n-a_m|= \frac{1}{n+1}-\frac{1}{m+1} \leq \frac{1}{n+1}+\frac{1}{m+1} \leq \frac{2}{N+1} < \epsilon $$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$B\in\Bbb R$ has the property that given $b∈B$ there exists $k>0$ such that if $0<|b−x|A subset $B$ of $\mathbb R$ has the property that given $b ∈ B$ there exists $k > 0$ such that if $0 < |b − x| < k$ for some $x ∈ \mathbb R$, then $x \notin B$. Is $B$ countable?
I tried using the diagonal argument here to show that ... | The property implies that for every $b \in B$ there exists $k_b > 0$ such that $\langle b - k_b, b + k_b\rangle \cap B = \{b\}$.
Pick a rational number $q_b \in \left\langle b - \frac{k_b}2, b + \frac{k_b}2\right\rangle$ and consider the map $f : B \to \mathbb{Q}$ given by $f(b) = q_b$.
Take $b,c \in B$ such that $q_b ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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$f(x)$ such that $f(x)*f(f(x)+\frac{1}{x})=1$ and $f(x) > - \frac{1}{x}$ $ \forall$ $ x > 0$ and $f(x)$ is an injective function I don't even know where to begin in solving this functional equation. I got this in a multiple choice question exam and was able to solve it by substituting all of the given options into the ... | Let's for a moment assume that $f(x)$ is invertible with $g(x)$ being its inverse. We can always discard our assumption later on if things don't turn out the way we want them to.
So, $f(x)f\biggl(f(x) + \frac{1}{x}\biggr) = 1$ can be modified as $x.f\biggl(x + \frac{1}{g(x)}\biggr) = 1$
$$\Rightarrow x + \frac{1}{g(x)}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Iwasawa decomposition of inverse Let $G$ be a semisimple rank one Lie group with finite center. Let $G=KAN$ be the Iwasawa decomposition with $\mathfrak{a}=$Lie($A)=\text{span}\ H$. Then if $G\ni g=kan, a=exp(tH)$ is it true that $$g^{-1}=\tilde k exp(-tH) \tilde n$$ is the decomposition of $g^{-1}$ where $\tilde k\in... | The answer is no. Consider $G=SL(2,\mathbb{R})$. Let
$$
g=kan=
\left(\begin{matrix}
0 & -1 \\
1 & 0
\end{matrix}\right) \cdot
\left(\begin{matrix}
1 & 0 \\
0 & 1
\end{matrix}\right) \cdot
\left(\begin{matrix}
1 & 1 \\
0 & 1
\end{matrix}\right) =
\left(\begin{matrix}
0 & -1 \\
1 & 1
\end{matrix}\right).
$$
We can write ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Garden with mushrooms A farmer cultivates mushrooms in his garden. A greedy neighbor wants to pick some but the farmer is trying to block him.
The garden has the form of a 8x6 grid. Rows 1 to 8 from the front to the back and columns A to F from left to right. The mushrooms are planted in the 8th row (6 mushrooms). The ... | The neighbor can win. He starts to E1 and claims the distant opposition. If the farmer moves forward, so does the neighbor. If the farmer moves sideways the neighbor moves diagonally forward on the side away from the farmer. Now the farmer must move toward the neighbor and the neighbor can move in front of the farm... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2768088",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Solving $z^3+3i\overline{z}=0$ $$z^3+3i\overline{z}=0$$
$z=x+yi$
$$(x+yi)^3+3i(x-yi)=0$$
$$x^3+3x^2yi-3xy^2-y^3i+3ix+3y=0$$
$$x^3-3xy^2+3y=0\text{ and } 3x^2y-y^3+3x=0$$
How to continue from here?
| Multiplying by $z$,
$$z^4+3i|z|^2=0$$ so that $z^4$ is purely imaginary. Then with $\omega$ a fourth root of $-i$,
$$-ir^4+3ir^2=0.$$
We have $r=0\lor r=\sqrt3$ and
$$z=0\lor z=\sqrt 3\,\omega$$ (five solutions in total).
| {
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Functional Analysis integral problem Im attempting to solve an example from my course,
Prove that for any $0<p<1$ there is a sequence of functions $(f_{i})_{i=1}^{\infty}$ in $C^{\infty}_{0}$($[-1,1]$) such that,
$$ lim_{i \to \infty}=\int_{\mathbb{R}} |\frac{df_{i}}{dx}(y)|^{p}dy=0 $$
and for any $x \in (-1,1) $ we... | Let $\phi: \mathbb R^+ \to [0,1]$, smooth, $\epsilon>0$
*
*$\phi(x) = 0$ for all x $\leq \epsilon$,
*$\phi(x) = 1$ for all $x \geq 1$ and
*$\phi'(x) \leq 2$ for all $x\in \mathbb{R}^+$.
For $x\in [-1,1]$ we now define
$$
f_i(x) := \phi(i (1-x^2)).
$$
Now, we only need to check the requested properties. Clearly... | {
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Is $G$ isomorphic to $\frac{G}{H}\times H$? Let $G$ be a group and $H$ a normal subgroup. Is $G$ necessarily isomorphic to $\frac{G}{H}\times H$?
| Not only need they not be isomorphic, but $G$ need not even have a subgroup (much less a direct factor) isomorphic to $G/H$. For example, $SL(2,5)$ has a center $Z$ of order 2, and $SL(2,5)/Z$ is isomorphic to $A_5$, but $SL(2,5)$ has no subgroup isomorphic to $A_5$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2768685",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 4
} |
estimation of the rest of Taylor expansion for holomorphic function let $f$ be a holomorphic function on $D=\mathbb{D}(0,1)$
Let $f(z)=\sum \frac{f^{(n)}(0)}{n!}z^n$ be its Taylor expansion.
If I use the Taylor McLaurin inequality:
$||f(z) - f(0) +f'(0)z|| \le Mz^2/2 $ where $M=Sup ||f''(z')||$ for $z' \in D(0,z)$?
Is... | Well, you need an absolute value on the $z$ ($z$ could be negative, of course). The integral form of the remainder remains valid,
$$ f(z) - \sum_{k=0}^n \frac{f^{(k)}(a)}{k!}(z-a)^k = \frac{1}{n!} \int_a^z (z-w)^n f^{(n+1)}(w) \, dw, $$
where the path can be chosen to be the line segment joining $a$ to $z$, but need no... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2768877",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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} |
Maclaurin's series for $\cos{\sqrt{x}}$ Why does Maclaurin's series for this function exist:
$$y=\cos{\sqrt{x}}$$
even though the second derivative of the function at $x=0$ is undefined?
I know that you can use the standard series of $\cos{x}$ and replacing $x$ with $\sqrt{x}$ to find its Maclaurin's expansion, but why... | You seem to misinterpret the nature of the chain rule $$(f\circ g)'(x) = f'(g(x))g'(x).$$ What it means is that if $g$ is differentiable at $x_0$ and $f$ is differentiable at $g(x_0)$, then $f\circ g$ is differentiable at $x_0$ and the derivative is given by the above formula.
Basically, a crude way to say it is "condi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2768961",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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In an $n$-dimensional manifold, can we say it has $n-k$ compact and $k$ noncompact dimensions? In theoretical physics it is quite common to talk about the number of compact and non-compact dimensions of a manifold, or even about compactified dimensions, and indeed often it is quite clear what is meant, like in a Cartes... | As Lee Mosher pointed out, it's probably not possible to come up with a completely general definition of the "number of compact dimensions" in an arbitrary manifold. But I think what physicists generally mean when they say a manifold $M$ has "$n−k$ compact dimensions and $k$ noncompact dimensions" is that $M$ is diffeo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2769086",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Buffon's Needle. Probability to intersect the line.
Problem: A table is ruled by equidistant parallel lines, 1 inch apart. A needle
of length 1 inch is tossed at random on the table. What is the
probability that it intersects a line?
Let's describe the distance from the needle-center to the nearest line by $D$ an... | The fun part of this is apparently that there are different interpretations which can be put on "random" although the one suggested seems reasonable to me. Needless to say, these give different answers.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2769573",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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How to find the key matrix of a 2x2 Hill Cipher?
In the english language, the most common digraph is TH which is then followed by HE. In this particular example let's say the digraphs with the most frequencies are RH and NI. How would I find the a, b, c, and d values for the key matrix:
\begin{pmatrix}a&b\\c&d\end{p... | You assume that $TH \to RH$ and $HE \to NI$ under the Hill cipher.
Or in matrix notation:
$$\begin{bmatrix} a & b\\ c & d\end{bmatrix}\begin{bmatrix}19\\7 \end{bmatrix}= \begin{bmatrix}17\\7 \end{bmatrix}$$
and
$$\begin{bmatrix} a & b\\ c & d\end{bmatrix}\begin{bmatrix}7\\4 \end{bmatrix}= \begin{bmatrix}13\\8 \end{b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2769737",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is this subspace of $\ell_\infty$ a Banach space? Define $T: \ell_\infty \rightarrow \ell_\infty$ the continuous linear operator defined by $$T(x_1,x_2,x_3,\dots) = (x_2,x_3,\dots).$$
Consider the subspace $M$ of $\ell_\infty$ defined by $$M = \{ x- T(x) : x \in \ell_\infty\}.$$
Is it true that $M$ is a closed subspace... | The sequence $(x_n)_{n\in\mathbb N}$ where $x_n=1/n$ is not in $M$.
In the other hand, the sequences $(x_n^m)_{n\in \mathbb N}$ where $x_n^m = x_n$ for $n\leq m$ and $x_n^m = 0$ else, are in $M$.
The convergence of $ (x_n^m)_{N \in \mathbb N}$ to $(x_n)_{n\in\mathbb N}$ in $\ell^\infty$ is clear.
In other words, $M$ i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2769973",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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I am looking at the implicit euler integration scheme for an equation $dx/dt = xt + 1$
How do you arrive at the second line from the first?
| I don't think the second line does follow from the first, for with
$x_{k + 1} = x_k + h(x_{k + 1} t_{k + 1} + 1), \tag 1$
we find
$x_{k + 1} = x_k + hx_{k + 1} t_{k + 1} + h, \tag 2$
whence
$x_{k + 1} - hx_{k + 1} t_{k + 1} = x_k + h, \tag 3$
or
$x_{k + 1}(1 - ht_{k + 1}) = x_k + h; \tag 4$
thus, provided $1 - ht_{k +... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2770107",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$I(x_1, x_2, . . . , x_6) = 2x_1 + 4x_2 + 6x_3 + 8x_4 + 10x_5 + 12x_6 \mod 10$ is the invariant
Each term in a sequence $1, 0, 1, 0, 1, 0, . . .$ starting with the seventh is the sum of the last $6$ terms $\mod 10$. Prove that the sequence $. . . , 0, 1, 0, 1, 0, 1, . . .$ never occurs.
Why $I(x_1, x_2, . . . , x_6) ... | So you have a recurrence and you want an invariant which distinguishes $1,0,1,0,1,0$ from $0,1,0,1,0,1$, so your invariant will have to take into account six consecutive terms, and can't just be a variety of the sum, because the sum for both sequences is $3$.
You have $x_{n+1}=x_n+x_{n-1}+x_{n-2}+x_{n-3}+x_{n-4}+x_{n-5... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2770242",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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First variation of volume in $\mathbb{R}^3$ Let $U \subset \mathbb{R}^3$ be a bounded subset with smooth boundary.
Let $Y \colon \mathbb{R}^3 \to \mathbb{R}^3 $ be a smooth vector field. I know that the first variation of the volume of $U$ w.r.t. $Y$ is given by
$$
\delta_Y|U| = \int_U \text{div}Y \, dx_1 dx_2dx_3.
$$
... | Let $\{\phi_t\}_{t\in (-\epsilon, \epsilon)}$ be the flow corresponding to $Y$. Then
\begin{align}
\delta_Y [U] &= \frac{d}{dt}\bigg|_{t=0} \int_{\phi_t(U)} \mathrm d x_1\mathrm d x_2\mathrm d x_3\\
&= \frac{d}{dt}\bigg|_{t=0} \int_{U} \phi_t^*(\mathrm d x_1\mathrm d x_2\mathrm d x_3) \\
&= \frac{d}{dt}\bigg|_{t=0} \i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2770371",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
The number of imaginary roots of $\sum_{n=1}^{100} \frac {n^2}{x-x_{n} }= 101$, where each $x_n$ is real
Determine the number of imaginary roots of the equation
$$\sum_{n=1}^{100} \frac {n^2}{x-x_{n} }= 101$$
where $x_{1}$, $x_{2}$, $x_{3}$, $\ldots$ are all real.
I did this question a few months back, but I am not a... | I think the question is ambiguous about whether it asks for purely imaginary roots or merely for complex roots that are not real.
But let's take a look to see what kind of roots we can find.
Let $x$ be a complex number. Suppose the imaginary part of $x$ is positive.
For $n$ an integer and $x_n$ real,
what can you say a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2770582",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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} |
Find the matrix of the linear maps $L : \mathbb{R}^3\to\mathbb{R}^3$ , where $n = (n1_, n_2, n_3)$ and let $L(v) := n\times v$.
Find the matrix of the linear maps $L :\mathbb{R}^3\to\mathbb{R}^3$, where $n = (n_1, n_2, n_3)$ and let $L(v) := n\times v$.
Having trouble with this question, hope someone can help.
| Let $v = (c_1,c_2,c_3)$ be an arbitrary vector in $R^3$ then from definition of cross product $$Lv = n\times v = (n_2c_3-n_3c_2,n_3c_1-n_1c_3,n_1c_2-n_2c_1)$$
so then in particular for $v = (1,0,0),(0,1,0),(0,0,1)$ we have the following images under $L$
$$L(1,0,0) = n\times(1,0,0) = (0,n_3,-n_2)$$
$$L(0,1,0) = n\times(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2770667",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Integration of $\sec^4 x$ While practicing for the AP exam, I came across an integral that I found interesting and I attempted to do by hand: $$\int(\sec^4 x)\, dx$$
Eventually I got stuck, but here are the steps I took-
$$\int(\sec^4 x) \,dx$$
$$\int(\sec^2 x)(\sec x)(\sec x)\,dx$$
$$\int(\sec^2 x)(\sec x) (\frac{\ta... | For this problem in particular, I would automatically do what @TheIntegrator did, but in general we can find
$$I_n=\int\sec^nx\ dx$$
$$I_n=\int\sec^{n-2}x\sec^2x\ dx$$
$$I_n=\int(\tan^2x+1)^{n-2}\sec^2x\ dx$$
Applying the substitution $u=\tan x$ gives
$$I_n=\int(u^2+1)^{n-2}du$$
Assuming that $n\geq2$ is an even integ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2770760",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 6,
"answer_id": 4
} |
Expected difference from mean: is it always zero? The quantity
$$
\mathbf{E}(x-\mu)=\int (x-\mu)P(x) dx
$$
is equal to zero for symmetric probability density functions. What about the others?
| If $\mu = \mathbb{E}[X]$, then
$$
\mathbb{E}[X - \mu] = \mathbb{E}[X] - \mathbb{E}[\mu] = \mathbb{E}[X] - \mu = \mu - \mu = 0
$$
regardless of the skewness of $X$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2770914",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Does simple convergence imply local uniform convergence Consider $f_\nu:\Omega\subset\Bbb R^n\mapsto \Bbb R,\ \nu\in\Bbb N$ some functions
Local uniform convergence is when $\forall x\in\Omega$ there exists an open neighborhood $U_x\ni x$ of $\Omega$ such that $f$ converges uniformally in $U_x$
And my question is: does... | Consider $f_n(x) = x^n$ on $[0,1]$, which converges pointwise.
However, for any neighborhood of $x=1$, there is another point $x'$ such that $|f_n(x') - 0|$ is close to $1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2771007",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Clarification on rules regarding division of two equations This is going to seem like a highly obscure question because I've likely missed something fairly obvious. Apologies in advance.
Consider the following equations
$$2k+2mk=k$$
$$k+3mk=mk$$
which must be valid for all real values of $k$.
Now obviously the two equa... |
Now obviously the two equations are equivalent
They are indeed equivalent, so you can ignore the second equation, and focus on just the first one:
$$
2k+2mk=k \quad\iff\quad 2mk+2k-k=0 \quad\iff\quad (2m+1)k=0
$$
The latter can only hold true for all values of $\,k\,$ iff the LHS is the zero polynomial i.e.$\,2m+1=0\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2771089",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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If $ Ax=b $ has solution what about $ y $ in $ A^Ty=0 $ If $ Ax=b $ has solution what about $ y $ in $ A^Ty=0 $? ($A$ is a $m \times n$ matriz).
Do we have them $ y^Tx = 0 $ or $ y^Tb = 0 $ and why ?
Should I explain using only the fact that $ b \in R^m $ and $ x \in R^n $ ? Or there is something more to say?
| HINT
We can't conlude in general since we need to distinguish the cases, as for example
*
*if $m=n$ and $Ax=b$ has solution for every $b$ then $A^Ty=0 \iff y=0$
*if $m=n$ and $Ax=b$ has infinitely many solutions for some $b$ then $A^Ty=0$ has infinitely many solutions
*...and so on
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2771204",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Proving that this sequence is convergent Given $a_1 = 2$, and $ a_{n+1} = \frac{a_n+5}{4} $ for all $n > 1$ , is this sequence convergent? Give a formal proof in either case (converges or diverges).
Attempt: I do think this converges, but cannot say for sure.
$a_1 = 2$
$a_2 = \frac{7}{4}$
$a_3 = \frac{27}{16}$
$a_4 = ... | The formal proof that the OP asked for can be found by combining his work with the (partial) answers given by Abra001 and robjohn.
If the sequence converges, then the sequence $(b_n)$ defined by $b_n = a_{n+1}$ converges to the same limit. So if the limit exists, it can only be $5/3$, as we see in the answer provided b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2771447",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 8,
"answer_id": 4
} |
Common tangent to a circle & parabola.
I atempted it as:
Let $(h,k)$ be the point where common tangent occurs to both the curves. So slope at this point must be equall to both curves. Thus for parabola, $$\frac{dy}{dx}= \frac{2}{k}$$ ,at point $(h,k)$. Similarly for circle $$\frac {dy}{dx}=\frac{(3-h)}{k}$$. Equatin... | Say that tangent at $(t^2, 2t)$ on parabola is tangent to the circle. So the slope of this tangent is $$2y\frac{dy}{dx}=4\implies\frac{dy}{dx}=\frac{1}{t}$$
Equation of the tangent is $$(y-2t)=\frac{1}{t}(x-t^2) \\ \implies x-ty+t^2=0$$
Since this is the tangent to the circle the distance of this line from center of ci... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If $f'''(x)$ exists on an interval $[a,b]$, does that mean $f(x)$ is continuous on $[a,b]$? Does this follow trivially from the fact that differentiability implies continuity, and if $f'''(x)$ exists, then $f(x)$ is differentiable and therefore continuous?
| If $f'''$ exists over $[a,b]$, then $f''$ is differentiable over $[a,b]$, then $f'$ is differentiable over $[a,b]$, then $f$ is differentiable over $[a,b]$, then $f$ is continuous over $[a,b]$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2771722",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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GRE Cumulative addition problem. The following problem is quoted from Manhattan 5lb Book of GRE Practice Problems, 2016ed.
Question: Molly worked at an amusement park over the summer. Every two weeks she was paid according to the following schedule: at the end of 1st $2$ weeks, she received \$$160$. At the end of each... | I will give a brief explanation
In $2$ weeks she earned $=160$
In the next two weeks she earned $=1+160=161$
In the next two weeks she earned $=1+161+160=322$
In the next two weeks she earned $=1+161+160+322=644$
In the final two weeks she earned $=1+322+161+160+644=1288=>$ Note that $1288$ is the amount she earned in ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2771826",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How can I calculate this summation? $\sum_{x=60}^{100} {100\choose x} $? How can I calculate this summation?
$$\sum_{x=60}^{100} {100\choose x} $$ ?
I don't have idea how to calculate it, I tried to arrive at a probability expression of a random variable that is binomial ($Bin(n,p)$) but But I did not succeed.
| $$ \sum_{x=0}^{100} {100 \choose x} = {(2)}^{100} $$
And using the fact that $ {n \choose r} = {n \choose n-r} $ it can be shown that $$ \sum_{x=50}^{100} {100 \choose x} = {(2)}^{99} $$
So $$ \sum_{x=60}^{100} {100 \choose x} = {(2)}^{99} - \sum_{x=50}^{59} {100 \choose x} $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2771933",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Confused by a solution given by professor $X$ and $Y$ are two continuous i.i.d random variables. They are both symmetric about zero. The problem is to show that
$P(|X+Y|<2|X|) > 0.5$
The model solution is the following:
$\iint\limits_{|x+y|<2|x|} f(x)f(y) \ dxdy = $
$\int_0^{\infty} (\int_{-3x}^x f(y) \ dy) \ f(x) dx \... | $f(-x)=f(x)$ and $f(-y)=f(y)$. Substitute $-x$ and $-y$ for $x$ and $y$ in the second integral, and it becomes the same as the first, which is where the $2$ comes from. Then just split the integral into two parts.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2772072",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Let $E \subset \mathbb{R}$ a null-set. Show that the subset $ \{(x,y) \in \mathbb{R}^2:x-y \in E \}$ is measurable Let $E \subset \mathbb{R}$ a null-set. Show that the subset $ \{(x,y) \in \mathbb{R}^2:x-y \in E \}$ is measurable.
We already know that if E is a $G_\delta $ set the statement is true. and we want to use ... | Consider the linear map $L:\Bbb R^2\to\Bbb R^2$, $L(x,y)=(x+y,y)$. Then, your set $S$ is $L(E\times \Bbb R)=\bigcup_{y\in\Bbb R}(E+y)\times\{y\}$. Since $S$ is image of a Lebesgue-measurable set by a Lipschitz-continuous homeomorphism, it is measurable as well.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2772185",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Let $_0, _1, _2, …$ be the sequence defined by the following recurrence relation: Let $_0, _1, _2, …$ be the sequence defined by the following recurrence
relation:
$_0 = 2$
$_1 = 2$
$_2 = 6$
$_ = 3_{−3}$ for $ ≥ 3$
Prove that is even for any nonnegative integer .
a. The base cases are = , = , and = .
$_ = , _ = ,$ ... | Even number is divisible by 2, so it is of the form $2i$ (not $2_i$) for some natural (or integer) $i$.
Probably easier to understand will be if you temporary forget induction. Because $a_0$, $a_1$, $a_2$ are even, their multiplications by 3 are also even, and their multiplication by 3 are also even, and so on. But it ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
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A field such that no extension has an 11th primitive root of unity I am asked to find the field described in the title. However, I can't quite understand the question.
For any field $K$, and $\zeta_{11} = e^{\frac{2\pi i}{11}}$ then surely the extension $K \leq K(\zeta_{11})$ contains the 11th primitive root of unity?... | Hint: what is the order of $\mathbb{F}_q^{\times}$, the roots of unity of a finite field $\mathbb{F}_q$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2772421",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why does the space of SPD matrices form a differentiable manifold? Disclaimer: I am not a mathematician, just a young neuroscientist trying to understand a paper. So forgive me if I have horribly misunderstood something.
So in order to understand this paper (that involves using a Riemannian kernel for a support vector ... | The set of symmetric matrices is a finite dimensional vector space. Any finite dimensional vector space is a manifold (just choose a basis to obtain a global chart). Let us denote the set of symmetric $n\times n$ matrices by $S$.
Now to be a positive definite matrix you will need that all the eigenvalues are positive. ... | {
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"url": "https://math.stackexchange.com/questions/2772654",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
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Finding coefficients in polynomial given three tangents I am stuck with a problem I simply cannot solve.
I have to find the coefficients of a quadratic polynomial given three tangents. The problem is stated as follows:
The three lines described by the equations
$y_1(x)=-4x-16.5$
$y_2(x)=2x-4.5$
$y_3(x)=6x-16.5$
are all... | Note that if a line and a quadratic are tangent $mx+d=ax^2+bx+c$ then the following quadratic will have discriminant zero
\begin{eqnarray*}
ax^2+(b-m)x+c-d=0.
\end{eqnarray*}
This will lead to $3$ equations for $a,b,c$ that are easily solved giving
$(a,b,c)=(1/2,1,-4)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2772749",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Berry-Esseen Smoothing Inequality from Feller Volume 2; Lemma 2 XVI.4 Fellow Feller fans,
I have a question concerning Fellers (An introduction to probability and its Applications, Volume 2 1971) treatment of the Berry-Esseen inequality obtained by smoothing which is Lemma 2 on page 538.
Specifically it is about going ... | It is precisely the Riemann-Lebesgue Lemma which will reduce the constant.
The exponential factor may not converge alone, but the whole integral vanishes:
$$
\int \frac{\mathrm{e}^{-i \zeta a}}{-i\zeta}\cdot (C_X(\zeta)-C_Y(\zeta))C_Z(\zeta)\mbox{d}\zeta = \int {\mathrm{e}^{-i \zeta a}}\cdot \frac{C_X(\zeta)-C_Y(\zeta)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2772819",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Find a basis and the dimension of the subspace R^3 consisting of all vectors of x Let $T_A : R^4 \rightarrow R^2$ be a multiplication by A. Find a basis and the dimension of the subspace $R^3$ consisting of all vectors of x for which $T_A(x)=0$ where A=
$$
\begin{bmatrix}
4 & 2 & 1 & -1 \\
7 & -1 & 0 & 2 \\
\end{bmatr... | Continuing on from @Math1000's very helpful comment it is now apparent that
$$\operatorname{null}T = \{(x_1,x_2,\frac{1}{2}x_1+\frac{5}{2}x_2,-\frac{7}{2} x_1+\frac{1}{2}x_2)\ |\ x_1,x_2\in\mathbf{R}\}$$ $\beta = \{(1,0,\frac{1}{2},-\frac{7}{2}),(0,1,\frac{5}{2},\frac{1}{2})\}\subseteq \operatorname{null}T$, and that ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2772914",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
R-linear categories Wikipedia "Preadditive category" and nLab "algebroid" seem to define $R$-linear category as one whose hom-sets are $R$-modules, where those modules are over a COMMUTATIVE ring. Does anybody knows why they require the ring to be commutative? When defining preadditive categories, categories whose hom-... | The source of the reason is in module theory. For $R$-modules $M$ and $N$, the set $\mathrm{Hom}_R(M,N)$ is naturally an abelian group. You'd prefer it to be an $R$-module since that's the category from which $M$ and $N$ came. This is true if $R$ is commutative. In particular, if $M$ is an $(R,S)$-bimodule then
$\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2773014",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
How to compute $\int_0^2 f(4x)\, dx$ given $\int_0^8 f(x)\, dx=4$
Compute $\displaystyle \int_0^2 f(4x)\, dx$ given that $\displaystyle\int_0^8 f(x)\, dx=4$.
At first I thought this was an ‘integrate by recognition’ type of question, I but can’t seem to come up with an answer.
Can someone tell me what sort of method ... | Let $u=4x$, then you get $$\int_0^2 f(4x)\, dx= (1/4)\int_0^8 f(u)\, du=1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2773111",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Showing $\sum x_n\sin(nx)$ converges uniformly
Show the $\sum\limits_{n = 1}^\infty x_n\sin(nx)$ converges uniformly iff $nx_n →0$ as $n →\infty$, where $x_n$ is a decreasing sequence and with $x_n>0$ for all $n=1,2,\cdots$.
My attempt was with M-test. I took $M_n= nx_n$. Since $\sin(x)$ is bounded by $1$, I claim th... | You cannot do this by directly applying some test for convergence. The proof involves 'summation by parts' and some calculations involving $\sin (nx)$ which are basic to Fourier series. A complete proof can be found here: Theorem 7.2.2 (part 1), p. 112 of Fourier Series: A Modern Introduction by R. E. Edwards. [ If the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2773228",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Possible periods of the elements in $\mathbb{Z}_n$ I'm looking into different order finding algorithms, and something i often notice is that a specific order happens more often than other orders for elements
in the multiplicative group $\mathbb{Z}_n^*$.
For example in $\mathbb{Z}_{77}^*$. The order is often 30. I foun... | The simplest answer is when $n=p$, a prime.
Then the multiplicative group $\mathbb{Z}_p^*$ is cyclic of order $p-1$ and so the orders that occur are exactly the divisors of $p-1$. Moreover, if $d$ divides $p-1$, then there are exactly $\phi(d)$ elements of order $d$ in $\mathbb{Z}_p^*$.
Now take $n=pq$, where $p$ and $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2773356",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Find the convergence domain of the series $\sum_\limits{n=1}^{\infty}\frac{(-1)^{n-1}}{n3^n(x-5)^n}$
Exercise: Find the convergence domain of the series $$\sum_\limits{n=1}^{\infty}\frac{(-1)^{n-1}}{n3^n(x-5)^n}$$
To solve the series I used the comparison test then the Leibniz criteria:
$$\sum_\limits{n=1}^{\infty}\f... | The OP has asked, in comments beneath various answers as well as in the question itself, what's wrong with his procedure, so I'm going to address that only.
There seem to be two fundamental misconceptions. First, in the comparison step, it looks like you are saying that
$${1\over n3^n}\le 1\implies {(-1)^{n-1}\over n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2773568",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 3
} |
Explanation of function $|y| = x$ First time posting here.
Studying this book and this statement came across.
In the equation $|y| = x$, $y$ is not a function of $x$ because every nonnegative $x$-value has two $y$-values. For example, if $x = 3$, $|y| = 3$ has the solutions $y = 3$ and $y = -3$.
Huettenmueller, Rhonda... | Let observe that by definition $|y|=x \implies x\ge 0$ and
*
*$y\ge 0 \implies |y|=x \iff y=x$
*$y< 0 \implies |y|=x \iff -y=x \iff y=-x$
then we have two different function both defined for not negative $x$ values.
Plot of |y|=x
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2773676",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Can't solve matrix multiplication with vector I try to solve this:
$$
\begin{bmatrix} 1-m & 0 & 0 \\ 0 & 2-m & 1 \\
0 & 1 & 1-m\end{bmatrix}
\begin{bmatrix} h_{1} \\ h_{2} \\ h_{3} \end{bmatrix} = 0
$$
That's what i tried:
=>
$$ (1-m) *h_1 = 0 $$
$$ (2-m) * h_{2} + h_{3} = 0 $$
$$ h_{2} + (1-m) * h_{3} = 0 $$
=>
$$ ... | Clearly one solution is $0$ vector, that is $h_i=0$ for all $i$. If you want a solution different from the $0$ the determinant of a given matrix must be $0$. So $$(1-m)((2-m)(1-m)-1)=0$$
so you get $m_1=1$ or $m^2-3m+1=0 \implies m_{2,3} = {3\pm \sqrt{5}\over 2}$
However in the first case we get $h_2=h_3=0$ and $h_1\i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2773815",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
how to compare the large numbers without using calculator and/or without large calculation for the given two multiplication below, what is the relationship (>,< or =) .. between this two multiplication?
100210*90021 and
100021*90210
I can simply compare these two results by calculation of multiplication but it is too... | Note that
$$100210 \cdot90021=(100021+189)(90210-189)=\\=100021\cdot90210+189\cdot 90220-189\cdot 100021$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2773936",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Why did mathematician construct extended real number systems, $\mathbb R \cup\{+\infty,-\infty\}$? I know some properties cannot be defined with the real number system such as supremum of an unbounded set. but I want to know the philosophy behind this construction (extended real number system ($\mathbb R \cup\{+\infty,... | From a "philosophical point of view", one of the reasons to define the extended real numbers is because the numbers $\pm \infty$ quantify a number of numerical and geometric mathematical objects and notions and make things simpler overall. In other words, they didn't do it for the sake of philosophy, they did it for th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2774011",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
"answer_count": 4,
"answer_id": 2
} |
Why is $\sqrt{-i} = -(-1)^{\frac{3}{4}}$ I was doing some research to look for fun on the properties of $i$, the square root of $-1$, and it got me thinking about what the square root of $-i$ was, or the square root of $-1$. I found on wolfram alpha that it is $\sqrt{-i} = -(-1)^{\frac{3}{4}}$
, but I can't find an ex... | If two complex numbers are equal, if a+ bi= c+ di then the real and imaginary parts are equal, a= c and b= d. That is part of the definition of the "a+ bi" notation. Her you have $A^2- B^2+ 2ABi= 0+ i$. You must have the real parts $A^2- B^2= 0$ and imaginary part 2AB= 1.
Personally I would have used the "polar form... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2774099",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Prove that the given metric satisfies Einstein's equations in the vacuum For the Schwarschild metric
$$\mathrm{d}s^2=-\bigg(1-\frac{2GM}{r}\bigg) \mathrm{d}t^2 + \bigg(1-\frac{2GM}{r}\bigg)^{-1}\mathrm{d}r^2+r^2\mathrm{d}\theta^2 + r^2 \sin^2\theta \mathrm{d} \phi^2$$
prove that $\mathrm{d}s^2$ satisfies Einstein's equ... | The Einstein equation reduces to $R_{\mu \nu}=0$ when $T_{\mu \nu}=0$. To see this, set first $T_{\mu \nu}=0$ in the Einsten equation
\begin{equation}
R_{\mu \nu}-\frac{1}{2} R g_{\mu\nu}+\Lambda g_{\mu\nu} = 0
\end{equation}
then multiply it by $g^{\mu\nu}$ to get
\begin{equation}
g^{\mu\nu} R_{\mu \nu}-\frac{1}{2}g^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2774249",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Showing the BFGS Update Hi guys I was reading a paper and it was left as easy to show that (for $y_k,s_k \in \mathbb R^n$ and $B_k \in \mathbb R^{n \times n}$)
$$B_{k+1} = B_k + \frac{ y_k (y_k-B_k s_k)^T+ (y_k-B_k s_k) y_k^T }{y_k^Ts_k } -\frac{\langle y_k- B_k s_k, s_k \rangle}{(y_k^Ts_k)^2 } y_k y_k^T $$
implies... | The first update formula is not BFGS. It is DFP update. They are different algorithms, it is not possible to rewrite one to another. However, there is a certain symmetry between them: if you look at the DFP update for the inverse Hessian $H_k=B_k^{-1}$ in the link above, you may notice that the BFGS update for $B_k$ lo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2774416",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Continuity of functions of two variables Find the points where the function $f(x,y)$ is continuous where
$$f(x,y)=\left\{ \begin{array}{ll}
\frac{x^2\sin^2y}{x^2+2y^2} & \mbox{if $(x,y)\not=(0,0)$};\\
0 & \mbox{if $(x,y)=(0,0)$}.\end{array} \right.$$
What I attempted: Here $f(x,y)$ is continuous at all the points $(x... | Let $f(x)=\sin^2 x$ and $g(x)=x$. Observe $f(0)=g(0)=0$. Furthermore, $f'(x)=2\sin x \cos x=\sin(2x)$ and $g'(x)=1$. But $f'(x)=\sin(2x) \leq 1=g'(x)$. Therefore, $f(x) \leq g(x)$ for $x \geq 0$. So your inequality is correct.
But this is making things too difficult. This is a Squeeze Theorem problem 'in reverse', mea... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2774564",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
A code that is systematic on every set of k coordinates I have the following definition:
Let $C$ be an $\mathbb {F} _{p}-[n, k, d]$ code. We say that $C$ is systematic on a set of $k$ coordinates $\{i_1, ..., i_k\}$ if there is exactly one codeword $c = (c_{i_{1}}, ..., c_{i_{k}}) \in \mathbb {F}_{p}^k$.
I am struggli... | It means that for any $k$ coordinates out of $n,$ and there are $\binom{n}{k}$ of those, those coordinates could conceivably serve as the original message coordinates. So it's a strong equidistribution property.
In other words, take the code $C$ apply any permutation $\sigma$ in $S_n$ to its coordinates, to obtain a ne... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2774900",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
counting measure space is not separable but the corresponding $L^p$ space is separable Let $\left(X,\mathcal{F},\mu\right)$ be a measure space. We define a pseudometric on measure space: for any $A,B\in\mathcal{F}$ $$d_{\mu}\left(A,B\right)=\mu\left(A\Delta B\right)=\mu\left(\left(A\setminus B\right)\cup\left(B\setminu... | The counting measure is separable. A measure space is separable iff it is generated by a countable collection of sets, modulo completion. In this case, the singletons generate the $\sigma$-algebra that is the power set.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2775036",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
How to optimize $\sum_{i=1}^{T}\|A_ix - b_i\|^2$ subject t0 $\sum_i x = 1$ and $x \geq 0$ How to solve the follow optimization problem?
$$\begin{array}{ll} \text{minimize} &\displaystyle\sum_{i=1}^{T} \| \mathbf{A}_i\mathbf{x} - \mathbf{b}_i \|^2\\ \text{subject to} & \mathbb{1}^\top {\mathbf{x}} = 1\\ & \mathbf{x} \ge... | I assume $\| \cdot \|$ is the $\ell_2$-norm. Let
$$
A = \begin{bmatrix} A_1 \\ A_2 \\ \vdots \\A_T \end{bmatrix},
\qquad
b = \begin{bmatrix} b_1 \\ b_2 \\ \vdots \\ b_T \end{bmatrix}.
$$
Then
$$
\sum_{i=1}^T \| A_i x - b_i \|^2 = \| Ax - b \|^2.
$$
So you can use the solution in the question you linked to.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2775125",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
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