Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Prove $\ \frac{z^{2010} - \bar z^{2010}}{1+z\bar z}$ is imaginary number Prove $\ \frac{z^{2010} - \bar z^{2010}}{1+z\bar z}$ is imaginary number.
I understand that if
$\ z = (a+bi) $ then $\ z - \bar z = 2bi $ and the denominator $\ 1+z\bar z $ is $\ 1+|z|^2 $ and therefore it is a real number. so need to prove the n... | $z\bar z=|z|^2$
$(a+ib)^n-(a-ib)^n=2i\sum_{r=0}^{2r+1\le n}\binom n{2r+1}a^{n-(2r+1)}b^{2r+1}(-1)^r$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2900911",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 3
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Find when $2^{3^{4^{...^{n}}}} \equiv 1$ (mod $n+1$) is true It is linked to my previous question, I haven't been given any clue for how to verify this modular equation:
$2^{3^{4^{...^{n}}}} \equiv 1$ (mod $n+1$)
How can I find the condition for $n$?
| $2^d \equiv 1 \mod (n+1)$ iff $n$ is even and $d$ is divisible by the multiplicative order of $2$ mod $(n+1)$. So for the condition to be true, you need the multiplicative order of $2$ mod $(n+1)$ to be a power of $3$. The first few $n$ for which this is the case are
$6, 72, 486, 510, 2592, 3408, 18150, 35550, 39366,... | {
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"url": "https://math.stackexchange.com/questions/2901004",
"timestamp": "2023-03-29T00:00:00",
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Mahlo operation, consistency border Can a (relatively consistent) cardinal notion be given so that its usual Mahlo operation
is (probably at least) not consistent?
| Sure - "is a successor ordinal." For any (fine, uncountable) infinite cardinal $\kappa$, the set of limit cardinals below $\kappa$ is a club which avoids the set of successor ordinals.
I suspect, though, that this isn't really what you want. So in the opposite direction, let me observe:
There is no known, currently-co... | {
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"url": "https://math.stackexchange.com/questions/2901110",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Convergence of P-adic series How can I prove that $\sum_{i = 0}^\infty 2^i$ doesn't converge with respect to $|\cdot|_3$ without resorting to the fact that $\sum a_i$ converges wrt $|\cdot|_p$ if and only if $|a_i|_p \to 0$? Is it even possible?
| The only thing that $\sum_0^\infty 2^n$ can converge to is $-1$. But
this series does not converge $3$-adically to $-1$. If it did then
$$\left|-1-\sum_{n=0}^N 2^n\right|_3<1$$
for all large enough $N$. This means that
$$\sum_{n=0}^N 2^n\equiv-1\pmod 3\tag{*}$$
for all large enough $N$. But $3\nmid 2^n$ for all $n$, so... | {
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"timestamp": "2023-03-29T00:00:00",
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Example of a metric on $\mathbb{R}^2$ that is not induced by any norm. From looking at the properties of both what makes a function a metric and a function a norm, I'd gather that I'd have to create a metric that would not satisfy the scalar multiplication property of a norm (i.e. $||ax|| \ne |a|||x||$). So, I went wit... | Yes, this is correct. All metrics induced by a norm have to be homogeneous (scalar property) and hence counterexamples only work if they are not homogeneous themselves. Your example works just fine, you just need to verify the triangle inequality which works out nicely.
Also, the discrete metric is a standard example f... | {
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Let $\lambda\in \sigma(A).$ Then, $e^{\lambda t}\in \sigma(e^{tA})$ where $\sigma(A)$ is set of all eigenvalues of matrix $A$ Let $\lambda\in \sigma(A).$ Then, $e^{\lambda t}\in \sigma(e^{tA})$ where $\sigma(A)$ is set of all eigenvalues of matrix $A$.
MY TRIAL
Let $\lambda\in \sigma(A).$ Then, $\exists:t\neq 0$ s.t.
... | Your idea is right but your proof is muddled (due to mixing up the scalar $t$ with the eigenvector of $A$ corresponding to $\lambda$).
Suppose $v$ is an eigenvector with $Av = \lambda v$. Then
$$e^{At}v = \sum_{i=0}^{\infty} \frac{(At)^i}{i!}v = \sum_{i=0}^{\infty} \frac{t^i}{i!} (A^i v) = \sum_{i=0}^{\infty} \frac{(\l... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Conflicting sign of line intergrals I am studying vector analysis (Mathematical Methods by Hassani). There is a passage in the book that talks about how parameterisation ensures that a line integral will have the correct sign.
What are the conditions that make a non-parameterised line integral have a different sign tha... | It seems that the Author is aimed to make the reader aware about the risk to proceed by coordinates, in that case assuming
*
*$\vec A=(A_x,A_y,A_z)$ with $A_x>0$
clearly along path $(iv)$
$$\vec A \cdot d\vec r=-A_xdx$$
and it leads to the wrong result
$$\int_{(a,a)}^{(0,0)} \vec A \cdot d\vec r=\int_{a}^{0} -A_xd... | {
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Domain for index of Radical Sign What is the domain of $\sqrt[x]{a}$, and is $\sqrt[x]{a}=a^{1/x}$ always true??
I was told that the domain of $\sqrt[x]{a}$ is natural numbers and the domain of $a^{1/x}$ is real numbers, so they are not identical. Is it true??
| Were you also told to not split infinitives and that sentences shouldn't end with prepositions like "by" or "with"?
By convention we agree that $\root x \of a$ refers to the principal $x$th root of $a$. If $x$ is a positive real integer, then $a$ has precisely $x$ roots, and each of the other roots can be obtained by m... | {
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"timestamp": "2023-03-29T00:00:00",
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Find all skew-symmetric matrices given their anti-commutator with a symmetric matrix Let $S$ be a skew-symmetric matrix and $J$ a symmetric matrix. Is it possible to find all skew-symmetric matrices $\Omega$ satisfying $$S = J\Omega + \Omega J$$ in terms of $S$ and $J$?
| Vectorizing the equation yields
$$\eqalign{
{\rm vec}(S) &= {\rm vec}(J\Omega I) + {\rm vec}(I\Omega J) \cr
s &= (I\otimes J + J\otimes I)\,\omega \,\,{\dot =}\,\, M\omega \cr
\omega &= M^+s + (I-M^+M)\,a \cr
\Omega &= {\rm Mat}\big(M^+s + (I-M^+M)\,a\big) \cr
}$$ where ${\rm Mat}()$ is the inverse of the ${\rm vec}()$... | {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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Tough Divisibility Problem
When the five digit number $2A13B$ is divided by $19$, the remainder is $12$. Determine the remainder of $3A21B$ when divided by $19$.
$$2A13B \equiv 12 \pmod{19}$$
$$20000 + 1000A + 100 + 30 + B \equiv 12 \pmod{19}$$
$$ 5 + 12A + 5 + 11 + B \equiv 12 \pmod{19}$$
$$ 21+ 12A+ B \equiv 12 ... | \begin{align}30\,000+1\,000A+200+10+B\equiv x\pmod{19}&\iff-1+12A+10+10+B\equiv x\pmod{19}\\&\iff12A+B\equiv x\pmod{19}.\end{align}Therefore, since $12A+B+9\equiv0\pmod{19}$, take $x=10$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving homeomorphism on closed ball Let $B$ be the closed unit ball in $\mathbb{R^n}$, and $a\in B,|a|<1$. Prove that $f:B\rightarrow B$,
$$f(x) = (1-|x|)a+x$$
Is an homeomorphism.
To me it's clear $f$ is continuous and I managed to prove it's injective. I'm only having trouble proving it's surjective, or finding an i... | This answer uses some light algebraic topology (really, just the definitions of homotopy, contractible) to show that $f$ is surjective. From there, showing $f$ is a homeomorphisn is straightforward.
Note first that $f$ preserves the unit circle (or, if you prefer, the boundary of the closed ball, i.e. its difference wi... | {
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"url": "https://math.stackexchange.com/questions/2901905",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Show that the number of elements of a finite set is well-defined. Given the following definition:
A set $A$ is finite if it is empty or there are $n\in \mathbb{N}$ and a 1-1 onto function $f:\{1,...,n\}\to A$. In the first case we say that $A$ has $0$ elements, while in the second we say $A$ has $n$ elements. We say th... | Sketch:
Hopefully you know that if $n\ne k$, then one of $\{1,2,\ldots,n\}$ and $\{1,2,\ldots, k\}$ is a proper subset of the other.
Now prove by induction on $n$ that a function from $\{1,2,\ldots,n\}$ to a proper subset of itself cannot be injective.
| {
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Does there exist a function which equals $0$ for odd inputs and $1$ for even inputs? Suppose $f(n)$ is a function that equals $0$ for odd inputs of $n$ and $1$ for even inputs. Note that $n$ can only be an integer. Is there a way of explicitly defining $f(n)$ so that satisfy the above conditions, without having to use ... | Another option is $$f(n) = \cos^2 \left(\frac{n\pi}{2}\right)$$
| {
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Find a sequence converging to $\{1/k\}^{\infty}_{k=1}$ in $l^2$ Let $s=\{1/k\}^{\infty}_{k=1}$. Find a sequence $\{s_n\}^{\infty}_{n=1}$ of points in $l^2$ such that each $s_n$ is distinct from $s$ and such that $\{s_n\}^{\infty}_{n=1}$ converges to $s$ in $l^2$.
This is a problem from Goldberg. Exercise 4.3, 3. I don'... | Hint. Recall that $l^2$ is the normed space of all square summable sequences $s$, such that
$$\|s\|^2_2=\sum_{k=1}^{\infty}s(k)^2<+\infty.$$
Let $s_n(k):=\frac{1}{k}+\frac{1}{nk}$ (also $s_n(k):=\frac{a_n}{k}$ with $a_n\to 1$ will work), then $\{s_n(k)\}_{k\geq1}\in l^2$ for each $n\geq 1$. Now consider the limit of
... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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if $\ A^2 = -I $ then $\ A $ has no real eigenvalues Given $\ A $ is a $\ 2 \times 2 $ matrix over $\ \mathbf R$ that $\ A^2 = -I $ and I need to prove $\ A $ has no real eigen values.
$\ A^2 = -I \rightarrow A^2 +I = 0 $
I guess it is something about $\ x^2 +1 = 0 $ has no real solutions... but if someone can show me ... | Since $A^2+I = 0$, the polynomial $x^2+1$ annihilates the matrix $A$. Therefore the minimal polynomial $m_A$ of $A$ divides $x^2 + 1$ so
$$\sigma(A) \subseteq \{\text{zeroes of } m_A\} \subseteq \{\text{zeroes of } x^2 + 1\} = \{i, -i\}$$
Hence $\sigma(A) \cap \mathbb{R} = \emptyset$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2902510",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 3
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Logic behind bitwise operators in C I came across bitwise operations in C programming, and I realized that XOR operator can be used to swap 2 numbers in their binary bases. For example let $$i=(65)_{10}=(1000001)_{2}, \text{ and } j=(120)_{10}=(1111000)_{2}$$.
Let $\oplus$ be the XOR operator, then observe that if I s... | Note that you can do the same thing without bitwise operators (at least for unsigned integer types since they can't overflow into undefined behavior):
// i == x j == y
i += j; // i == x+y j == y
j -= i; // i == x+y j == -x
i += j; // i == y j == -x
j = -j; // i == y j == x
Now if we do this bit... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2902731",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Evaluate: $ \int \frac{\sin x}{\sin x - \cos x} dx $ Consider
$$ \int \frac{\sin x}{\sin x - \cos x} dx $$
Well I tried taking integrand as $ \frac{\sin x - \cos x + \cos x}{\sin x - \cos x} $ so that it becomes,
$$ 1 + \frac{\cos x}{\sin x - \cos x} $$
But does not helps.
I want different techniques usable here.
| Let $$I_{1} = \int \frac{\sin x}{\sin x - \cos x}dx, \quad I_{2} = \int \frac{\cos x}{\sin x - \cos x}dx$$
Then $$I_{1}-I_{2} = \int 1\, dx = x + c_{1}$$
$$I_{1}+I_{2} = \int \frac{\sin x + \cos x}{\sin x - \cos x} dx = \ln(\sin x - \cos x) + c_{2} $$
Then solve simultaneously
$$I_{1} = \frac{1}{2}\left(x+ \ln(\sin... | {
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"timestamp": "2023-03-29T00:00:00",
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Finding $P(A \cap B)$, given $P(A\mid B)$ and $P(B\mid A)$ Given events $A$ and $B$, if $P(A\mid B)$ and $P(B\mid A)$ are known, can $P(A \cap B)$ be found? I tried the following approach and came to an answer, but doubt its veracity. Here's my attempt:
I tried thinking of $P(A\mid B)$ as $P(B \implies A)$. This is log... | Your assumption that $$p(A\mid B)=p(A \implies B)$$ is not valid.
Think of $A$ as $x>3$ and $B$ as $x>5$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2902945",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Finding Jordan canonical form of a matrix given the characteristic polynomial I am trying to find the Jordan canonical form of a matrix $A$ given its characteristic polynomial. Suppose $A$ is a complex $5\times 5$ matrix with minimal polynomial $X^5-X^3$. The end goal of the problem is to find the characteristic polyno... | We have the minimal polynomial is $X^3(X^2-1)$. Over $\Bbb C$, the exponent of the irreducible factor $(x-a)$ in the minimal polynomial gives the size of the largest Jodan block. Thus we have a $3\times 3$ Jordan block corresponding to the eigenvalue $0$. The only possibility is the middle one:
$$A=\left(
\begin{array}... | {
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"timestamp": "2023-03-29T00:00:00",
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Determining the general solution for the trigonometric equation $ 5\cos(x)-12\sin (x) = 13 $ Given that
$$5\cos(x)-12\sin (x) = 13 $$
I'm trying to evaluate the general solution for that expression. It reminds me of $5-12-13$ triangle. Since we don't know the degree of $x$, I couldn't proceed further. Specifically, let... | This has a general method: divide the whole equation by $\;\sqrt{5^2+12^2}=13\;$ , so the equation becomes
$$\frac5{13}\cos x-\frac{12}{13}\sin x=1$$
Since $\;\left(\frac5{13}\right)^2+\left(\frac{12}{13}\right)^2=1\;$ , there exists $\;\alpha\in\Bbb R\;$ (in fact, we can choose this value in an infinite number of ways... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Question about the definition of the least upper bound property Definition: Let $A$ the set with order relation. We say that the set $A$ has least upper bound property if any $A_0\subset A$, $A_0\neq \varnothing$ which has upper bound has the least upper bound.
Question 1: When we say "has upper bound..." do we mean t... | Your point is that if a set $A$ has the least upper bound property, it does not imply that every subset of A also has the least upper bound property.
Yes, you are quite right.
A good example is the set of rational numbers which does not have the least upper bound property while it is a subset of real numbers which ha... | {
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How can I find the limit of the following sequence $\sin ^2 (\pi \sqrt{n^2 + n})$? How can I find the limit of the following sequence:
$$\sin ^2 (\pi \sqrt{n^2 + n})$$
I feel that I will use the identity $$\sin ^2 (\pi \sqrt{n^2 + n}) = \frac{1}{2}(1- \cos(2 \pi \sqrt{n^2 + n})), $$
But then what? how can I deal with... | You can check $\sin^2(\pi\sqrt{n^2 + n})=\sin^2(\pi\sqrt{n^2 + n}-\pi n)$.
So $$\sin^2(\pi\sqrt{n^2 + n})=\sin^2 \pi\frac{n}{\sqrt{n^2 + n}+n}\to \sin^2\frac{\pi}{2}=1.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2903413",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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Prove that $\int_0^1\,\frac{\ln(x)}{\sqrt{1-x^2}}\,\text{d}x=-\frac{\pi}{2}\,\ln(2)$. I have discovered via contour integration that
$$\int_0^\infty\,\frac{\exp(t\,u)}{\exp(u)+1}\,\text{d}u={\text{csc}(\pi\,t)}\,\left(\frac{\pi}{2}-\int_0^{\frac{\pi}{2}}\,\frac{\sin\big((1-2t)\,y\big)}{\sin(y)}\,\text{d}y\right)\tag{*}... | \begin{align}
\int_0^1 x^\alpha\ dx &= \dfrac{x^{\alpha+1}}{\alpha+1}\\
\dfrac{d}{d\alpha}\int_0^1 x^\alpha\ dx &=\dfrac{d}{d\alpha}\dfrac{x^{\alpha+1}}{\alpha+1}\\
\int_0^1 x^\alpha\ln x\ dx &=\dfrac{x^{\alpha+1}((\alpha+1)\ln x-1)}{(\alpha+1)^2}\Big|_0^1=\dfrac{-1}{(\alpha+1)^2}
\end{align}
\begin{align}
I
&= \int_{0... | {
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"timestamp": "2023-03-29T00:00:00",
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Pell's equation (or a special case of a second order diophantine equation) Question Find integers $x,y$ such that $$x^2-119y^2=1.$$
So far I've tried computing the continued fraction of $\sqrt{119}$ to find the minimal solution, but either I messed up or I don't know where to stop computing a rough approximation of sai... | The algorithm in the first answer works for the general case, but in this specific case, noticing that $119$ is very close to the perfect square $121$ suggests that $y=11$ is worth a look. Then, $119\cdot 121=(120-1)(120+1)=120^2-1$ leads directly to $x=120, y=11$ as a solution.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Extension operator for Lipschitz domain in Sobolev spaces Suppose we have a Lipschitz domain $\Omega \subset \mathbb{R}^2$. Let $u$ be a function in the Sobolev space $W^{1\,,\,p}(\Omega)$. Since $\Omega$ is Lipschitz, there is an extension operator $P: W^{1\,,\,p}(\Omega) \to W^{1\,,\,p}(\mathbb{R}^2)$ such that
\beg... | Let us say that a domain $\Omega \subset \mathbb{R}^n$ is an extension domain if there exists a bounded linear operator $\mathcal{E} \colon W^{1,p}(\Omega) \to W^{1,p}(\mathbb{R}^n)$ such that $\mathcal{E}u(x) = u(x)$ for $x \in \Omega$.
It is proved in P. W. Jones, Quasiconformal mappings and extendability of function... | {
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Can the Inverse Finding the Laplace transform of $\frac{2s + 1}{s(s + 1)(s + 2)}$ without using partial fractions? I'm wondering if we can perhaps using the convolution theorem to find the inverse Laplace transform of $\dfrac{2s + 1}{s(s + 1)(s + 2)}$? I can find it using partial fraction decomposition, but it is not o... | It is not right away the convolution of two functions but you can split into two fractions and use convolution on each one and add the results .
| {
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"timestamp": "2023-03-29T00:00:00",
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Is the sequence $\frac{e^n}{n}$ convergent? Is the sequence $$\frac{e^n}{n}$$ convergent?
I think it is not because $\log n < n$, implying that $\frac{e^n}{n} >1$ and hence the limit does not exist. Which probably also means that the sequence is unbounded.
Am I right? Please correct me if I am wrong.
| No it is not. Notice that $$a_{n+1}=\dfrac{e^{n+1}}{n+1}=e\cdot \dfrac{e^n}{n}\cdot \dfrac{n}{n+1}>\dfrac{e}{2}a_n>1.3a_n$$therefore $$a_{n+1}>1.3a_n>(1.3)^2a_{n-1}>\cdots>(1.3)^na_1=e\cdot(1.3)^n$$which is unbounded since $e\cdot(1.3)^n$ is unbounded.
| {
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"url": "https://math.stackexchange.com/questions/2903890",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Smart Integration Tricks I am in the last year of my school
and studied integration this year
I have done several Integration techniques like Integration
*
*By substitution
*By partial fractions
*By parts
*Trigo. substitutions
*Tangent half angle substitution
and many other basic methods of integration.
So I... | Another neat trick is to add different forms of integrals to obtain a much simpler one.
For example, if we let a function $f$ be such that $f(x)f(-x)=1$ and we want to evaluate $$I=\int_{-1}^1\frac1{1+f(x)}\,dx$$ then we could replace $x$ by $-x$ giving $$I=-\int_1^{-1}\frac1{1+f(-x)}\,dx=\int_{-1}^1\frac{f(x)}{1+f(x)}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2903959",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 7,
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Does $\frac{n}{\sum\limits_{k=1}^{n}\Big(\frac{k}{k+1}\Big)^k}$ converge?
Does the sequence $$\displaystyle \frac{n}{\sum\limits_{k=1}^{n}\Big(\frac{k}{k+1}\Big)^k}$$
converge?
Attempt. Since $\Big(\frac{k}{k+1}\Big)^k \rightarrow 1/e\neq 0$ and the terms are positive, the series
$\sum\limits_{k=1}^{\infty}\Big(\fr... | If $a_n\to L,$ then as is well known, $(a_1+\cdots + a_n)/n \to L.$ Since $[n/(n+1)]^n \to 1/e,$ we therefore have
$$\frac{\sum_{k=1}^{n}[k/(k+1)]^k}{n} \to \frac{1}{e}.$$
Taking reciprocals gives the limit of $e.$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What you can conclude about signs of number $\lambda$ and $\omega$ Let matrix $A\in M_n(\mathbb R)$ is symetric matrix such that $a_{11}=-1,a_{22}=-1,a_{33}=1$ and spectrum of $A=\{\lambda,\omega\}$ such that $\lambda>\omega$. $N(A-\lambda I)=L(v_1,v_2)$ and $N(A-\omega I)=L(v_3)$ where is
$v_1=(1,1,0),v_2=(1,1,1),v_3... | Clearly $n=3$ because $A$ diagonalizes so the eigenvectors span the space.
We have $$N(A - \lambda I) = \operatorname{span}\left\{\pmatrix{1 \\ 1 \\ 0}, \pmatrix{1 \\ 1 \\ 1}\right\} = \operatorname{span}\left\{\frac1{\sqrt2}\pmatrix{1 \\ 1 \\ 0}, \pmatrix{0 \\ 0 \\ 1}\right\}$$
$$N(A - \omega I) = \operatorname{span}\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Triangles : isoceles and angles inside with other triangle I have this, what seems basic triangle problem, and not certain if the given information is sufficient to solve the problem of finding an angle.
We have one main isoceles and another one inside of it.
I have attached here a diagram, and we wish to find the angl... | Let $\widehat{QRP}= \widehat{RPQ}=x$, so $\widehat{RQP}=180-2x$, so $\widehat{RQS}=156-2x$, so the sum of the angles $\widehat{QST}$ and $ \widehat{QTS}$ is $24+2x$, so they are both $x+12$, so $\widehat{RTS}=168-x$, so $\widehat{RST}=12$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Forking but not dividing Definition. A formula $\phi(x,a)$ divides over a set $B$ if there are $k<\mathbb{N}$ and a sequence $(a_i)_{i<\omega}$ such that
(1) $\text{tp}(a/B)=\text{tp}(a_i/B)$, for all $i<\omega$;
(2) $\{\phi(x,a_i)\}_{i<\omega}$ is $k$-inconsistent.
Definition. A formula $\phi(x,a)$ forks over a se... | No, forking does not always imply dividing. In simple theories dividing and forking are the same, but they are not the same in general. Look at the following example.
Example. Let $\mathcal{L}=\{ R^{(3)} \}$ be a language which consists of a ternary relation. Consider the $\mathcal{L}$-structure $\mathcal{M}=\big(\mat... | {
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"source": "stackexchange",
"question_score": "3",
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How to find a bijective mapping such that a) $f: \mathbb{N} \rightarrow \mathbb{N} \cup \{0\}$
I'm thinking $f(x) = x-1$, because in order to get 0, it would just be 1-1. Is this correct? I can show this is a bijection because it is surjective, i.e.: $ y = x - 1 \implies x = y+ 1$ and $f(x) = f(y+1) = y+1-1 = y$ and it... | That would be correct. This is a variant of the Hilbert hotel "technique", whereby you can essentially lose an extra element in (and only in) an infinite set.
As Cantor said, "i see it but I can't believe it. "
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to factor polynomials in $\mathbb{Z}_{n}[x]$ I realize there already is a question almost identical to this (here), but the answers given are a bit too vague.
The problem I have is that I'm looking to factor the polynomial $$x^2+23x+18 \text{
in } \mathbb{Z}_{28}$$ into as many ways as possible. I've tried through... | Hint: Solve mod $4$ and mod $7$ and use the Chinese Remainder Theorem.
EDIT: OK, let's do the mod $4$ part. You want
$$ x^2 + 3 x + 2 \equiv (x+a)(x+b) \mod 4 $$
Thus
$$ \eqalign{a b &\equiv 2 \mod 4\cr
a + b &\equiv 3 \mod 4\cr} $$
One of $a$ and $b$, let's say $a$, must be even, but can't be $0$, so it... | {
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Find $f(10)=?$ when the following condition is given. $$ Let \ f:R \to R \in \vert f(x)-f(y) \vert \le (x-y)^3 \ \forall \ x,y \in R \\and \ f(2)=5;\ then\ f(10)=? $$
This question is from an old assignment on the topic Limits , Continuity and Differentiability
Though i didn't get the answer , but i tried in the foll... | Maybe, the inequality condition of the problem should be $$|f(x)-f(y)|\leq |x-y|^3,~~\forall x,y \in \mathbb{R}.$$
Thus, $$-|x-y|^2 \leq \frac{f(x)-f(y)}{x-y}\leq |x-y|^2, ~~~\forall x \neq y.$$
Now, fix $x$ and take all the limits under the process $y \to x.$ We may obtain $$0\leq\lim_{y \to x}\frac{f(x)-f(y)}{x-y} \l... | {
"language": "en",
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age-based word problem Peter's age is three years more than three times his son's age. After three years, Peter's age will be ten years more than twice his son's age. What is Peter's present age?
I have tried to put this into algebra, but not sure if correct?
$x =$ Peter's son's age
$p =$ Peter's age
\begin{align*... | The first equation ($3x + 3 = p$) is correct, but you lost your focus on the second equation. So now Peter's age is $p$ and his sons age is $x$. After three years, Peter's age will be $p+3$ and his son's $x+3$. So "After three years, Peter's age ($p+3$) will be ten years more than twice his son's age" becomes
$$
(p+3) ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is a density nonnegative almost everywhere? Let $(\Omega,\mathcal F,P)$ be a probability space. Let $\xi:\Omega\to\mathbb R$ be a random variable. Let $P_\xi$ be the distribution of $\xi$. Suppose $P_\xi$ is a continuous distribution. (i. e. There is a Borel function $f_\xi:\mathbb R\to\mathbb R$ such that for all $B\i... | What follows after "Assume $\mu(C)>0$" in your question should be followed by this:
$C=\bigcup_{n=1}^{\infty}C_n$ where $C_n:=f_{\xi}^{-1}((-\infty,-\frac1n])$ so that $C_1\subseteq C_2\subseteq\cdots$ and consequently $\mu(C_n)\uparrow\mu(C)$
Then assuming $\mu(C)>0$ leads to $\mu(C_n)>0$ for $n$ large enough and cons... | {
"language": "en",
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Show that $(a-b)(c-d)(\bar{a}-\bar{d})(\bar{c}-\bar{b})+i(c\bar{c}-d\bar{d})\text{Im}(c\bar{b}-c\bar{a}-a\bar{b})$ is real This is an exercise in Remmert's Theory of Complex Functions, GTM 122, page 17.
Show that for $a,b,c,d\in\mathbb{C}$ with $\lvert a\rvert=\lvert b\rvert=\lvert c\rvert$ the complex number $$(a-b)... | If $b=0$ or $c=0$, then the complex number equals $0$ which is real.
In the following, $b\not =0$ and $c\not=0$.
In order to prove that the claim is true, it is sufficient to prove the following two lemmas :
Lemma 1 : $$\text{Im}(c\bar{b}-c\bar{a}-a\bar{b})=-\frac{\bar a(a-b)(b-c)(c-a)}{2bc}$$
Lemma 2 : $$\text{Im}((a-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2904979",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Curvature function and rate of change of angle
Let $\gamma:(a,b)\rightarrow \mathbb{R}^2$ be a smooth curve with $\| \dot{\gamma}(s)\|=1$ for all $s\in (a,b)$. Fix $s_0\in (a,b)$ and let the unit vector $\dot{\gamma}(s_0)$ be represented by $(\cos \phi_0,\sin\phi_0)$. Then there is smooth function $\phi$ with $\phi(s... | Intuition is that you want to get $\phi$ by integration of $\kappa=\frac{d\phi}{ds}$.
You get $$\phi(s)=\phi_0 + \int_{s_0}^s \kappa(u) du$$
by fundamental theorem of calculus. From serret formulae you can derive $$\kappa=\langle \ddot \gamma,J \dot \gamma \rangle$$
where $\dot\gamma$ interpreted as tangential vector... | {
"language": "en",
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If $u$ and $w$ belongs to the same connected components, does there exist any $u-w$ path containing $v$?
If $u$ and $w$ belongs to the same connected components, does there exist any $u-w$ path containing $v$?
|
If $v$ is a cut vertex of $G$, then $G-v$ is disconnected and has at least two components , $G_1$ and $G_2$. Take $u \in G_1$ and $w \in G_2$.
You ask, “what happens if $u$ and $w$ lie in the same connected component of $G-v$?”
If $u$ and $w$ were in the same component of $G-v$, let's say $G_1$, there would be a pat... | {
"language": "en",
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Constant in front of the characteristic polynomial of a matrix In a text, I saw that the characteristic polynomial for an $n\times n$ matrix A with eigenvalues $e_{1}, \ldots e_{n}$ can be written
$$p(\lambda) = (\lambda - e_{1})(\lambda - e_{2}) \cdots (\lambda - e_{n}).$$
But shouldn't there be a constant in front of... | In principle it could happen if we knew nothing about the characteristic polynomial's structure.
However, it turns out that it will always be monic. One possible definiton for the determinant of a matrix is
$$
\det A = \sum_{\sigma \in \mathbb{S}_n}\operatorname{sgn}(\sigma) \cdot a_{1,\sigma(1)}\cdots a_{n,\sigma(n)}... | {
"language": "en",
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Degree of curve where matrix of polynomials has rank 1 My question is about a step in Exercise 12.8 on page 442 of 3264 & All That by Eisenbud and Harris. Chapter 12 is about Porteous' formula.
The exercise reads: Let $A=(P_{i,j})$ be a $2 \times 3$ matrix whose entries $P_{i,j}$ are general polynomials of degree $a_{... | I believe you got confused with the map. Let's define a bundle map by right multiplication with
$$
A = \begin{bmatrix}
P_{11} & P_{12} & P_{13} \\
P_{21} & P_{22} & P_{23}
\end{bmatrix}
$$
whose degrees are respectively$$ \begin{bmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{2... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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determinant and invertibility for matrices I'm reading this in my text:
I don't understand this at all: (see attachment)
It's the second part that I don't get. I get that if B has a row of 0s, then the determinant is 0 (I could cofactor expand this and it'd be 0). But here's what I don't get:
*
*if $|B| = 0$, why d... | Note that every elementary row operation is a multiplication of your matrix by an invertible matrix.
So if you can reduce your matrix to the identity matrix via elementary row operations that means you have multiplied your matrix by some invertible matrices to get $I$, therefore the product of those matrices is the in... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Zeroes of a polynomial. Evaluate an expression Let $x_1,x_2,x_3$ be the zeros of the polynomial $7x^3+24x^2+2016x+i$. Evaluate $(x_1^2+x_2^2)(x_2^2+x_3^2)(x_3^2+x_1^2)$.
My thoughts: I've tried $7(x-x_1)(x-x_2)(x-x_3)=0$ and expanded it out to match the polynomial given and got an ugly system of equations (which I can ... | Let $g(x)=b_3 x^3 + b_2 x^2 + b_1 x + b_0$ be a cubic having as zeros exactly $x_1^2+x_2^2$, $x_2^2+x_3^2$, $x_3^2+x_1^2$. Then $(x_1^2+x_2^2)(x_2^2+x_3^2)(x_3^2+x_1^2)=-b_0/b_3$.
Now $x_1^2+x_2^2=p_2-x_3^2$ etc. and so the roots of $g$ are given by $h(x_i)$ where $h(x)=p_2-x^2$. Note that $p_2=(x_1+x_2+x_3)^2-2(x_1 x_... | {
"language": "en",
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approximation to standard normal distribution I have a sequence of independent, but non-identically distributed Bernoulli random variables $X_i$'s taking on value $1$ with probability $p_{i1}\cdot p_{i2}$, where $i=1,\ldots,n$. Let $X=\sum_{i=1}^n X_i$. Using Lyapunov central limit theorem, I have approximated the pro... | Summary
Notation: Let $F_n$ be the cumulative distribution function for $X = \sum_{I=1}^n X_i$, and let $F$ be the cumulative distribution function $Z$ where $Z$ follows a standard normal distribution.
The result that you have only establishes that for any fixed $k$, $P[Z \geq k]$ will be an increasingly good approxim... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Why we get inequality form an equation? In the paper linear forms in the logarithms of real algebraic numbers close to 1, it is written on page 5 that-
$\varLambda \leq \frac{1}{by^n}$ (see equation 7 on page 5)
But we get it from an equation. As I understand , it should be $\varLambda = \frac{1}{by^n}$.
How $\va... | $$a=b$$ implies both $$a\le b$$ and $$a\ge b.$$
It's the author's choice to weaken the comparison for the requirements of the exposition.
Quiz:
Are these propositions true ? ($\land$ is and, $\lor$ is or)
*
*$a=b\implies a\le b\land a\ge b$ ?
*$a=b\implies a\le b\lor a\ge b$ ?
*$a=b\implies a< b\lor a=b\lor a> b$... | {
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Hilbert-Schmidt operator defined by non-orthogonal basis I have the following operator on $L^2(0,1)$
$$ Tf = \sum_{n \geq 0} 2^{-n}\langle f,v_n\rangle v_n$$
Where $v_n(t) = t^n$.
I was able to prove that it is Hilbert-Schmidt, but now I need to calculate its Hilbert-Schmidt norm, and its integral kernel. The problem i... | The operator is given by
$$
Tf(x)=\int_0^1 f(t) \sum_{n\ge 0} \frac{(xt)^n}{2^n}\, dt =\int_0^1 f(t)\frac{2}{2-xt}\, dt.$$
Let $K(t, x):=2(2-xt)^{-1}$. This is the kernel you are looking for. The squared Hilbert-Schmidt norm of $T$ is given by the integral
$$
\iint_{[0, 1]^2} \frac{4dtdx}{(2-xt)^2}= 2\log(2).$$
(th... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why is the product of elementary matrices necessarily invertible? Why is the product of elementary matrices necessarily invertible? I understand that each elementary matrix is invertible, but why is their product also invertible? Is it the indirect result of this theorem?
| That $A$ is invertible means precisely that there is another matrix $A^{-1}$ such that
$$AA^{-1}=I=A^{-1}A.$$
It is easy to show that the inverse, if it exists, must be unique.
Now suppose $A,B$ to be invertible, and denote $C:=AB$. $C$ will be invertible if we can find $C^{-1}$ such that $CC^{-1}=I=C^{-1}C$. Observe t... | {
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Why is it that $x\sim y$ implies that $[x]=[y]$? I had a doubt in the proof of this property:
For $x\in A$, let $[x]$ be the set $[x]=\{a\in A\mid a\sim x\}$.
a) $[x]=[y]$ or
b) $[x]\cap[y]=\varnothing$
I understand till the part where they prove that if for some $c$, $c\sim x$ and $c\sim y$ then $x\sim y$. But then... | If you understood that far, you're practically done. If $x\sim y$, then take $c\in[x]$, then $c\sim x$ and therefore $c\sim y$, since $x\sim y$, so $c\in[y]$. Therefore $[x]\subseteq[y]$. The argument for $[y]\subseteq[x]$ is similar, and therefore $[x]=[y]$.
| {
"language": "en",
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Does every finite non-trivial complete group have even order? Does every finite non-trivial complete group have even order?
I checked three well known classes of complete groups, and this statement is true for them all:
1) Symmetric groups:
All symmetric groups have even order (a well known fact)
2) Automorphism group... | No. There was at one time a conjecture that this is true, but an example of a complete group of order $3\cdot 19\cdot 7^{12}$ was produced by R.S. Dark in “A complete group of odd order”.
| {
"language": "en",
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Comparing Fisher Information of sample to that of statistic Let $X_1,...,X_n$ be Bernoulli($p$) where $p$ is unknown, and $n>2$, and let $T=X_1+X_2$. My task is to calculate the information about $p$ in the entire sample and compare it to the information about $p$ given by the statistic.
After a few lines of work, I o... | Fisher Information Matrix (FIM), is the negative of the Expectation of the Hessian of the log likelihood function, namely
\begin{equation}
I(p) = - E H(p)
\end{equation}
where $I(p)$ is the FIM and $H(p)$ is the Hessian. Your likelihood function for independent samples is
\begin{equation}
L(p) = f(x_1\vert p) \ldots ... | {
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Solving equation with fraction I don't understand how to get from the second to the third step in this equation:
$ - \frac { \sqrt { 2 - x ^ { 2 } } - x \left( \frac { - x } { \sqrt { 2 - x ^ { 2 } } } \right) } { \left( \sqrt { 2 - x ^ { 2 } } \right) ^ { 2 } } = - \frac { \sqrt { 2 - x ^ { 2 } } + \frac { x ^ { 2 } ... | It's simply that for $a>0$, $$\sqrt{a} = \sqrt{a}\cdot\underbrace{\left(\frac{\sqrt{a}}{\sqrt{a}}\right)}_1$$
$$=\frac{\sqrt{a}\cdot\sqrt{a}}{\sqrt{a}}$$
$$=\frac{(\sqrt{a})^2}{\sqrt{a}}$$
$$=\frac{a}{\sqrt{a}}$$
In your case, "$a$" is $2-x^2$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is the piecewise-defined function differentiable The function is defined as
$$f(x)=\begin{cases}x^2, &\text{ for }x\leq 1\\ \sqrt{x}, &\text{ for }x>1\end{cases}$$ and
Is this function differentiable at $x=1$?
I thought that since $\lim_{x\to 1}$ of $f'(x)$ exists then it IS differentiable. And I think this limit does... | Taking $\lim\limits_{x\to 1}f'(x)$ is not well defined, until we are convinced that $f'(x)$ exists. If $f'(1)$ does exist, we will have that the left and right hand limits defining the derivative are equal. If we denote $f'_-(x)$ and $f'_+(x)$ as the left and right derivatives respectively, we get that \begin{align*} f... | {
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Getting $p_y(y) = p_x(g^{-1}(y)) \left| \frac{\partial{x}}{\partial{y}} \right|$ by solving $| p_y(g(x)) \ dy | = | p_x (x) \ dx |$? My textbook has a very brief section that introduces some concepts from measure theory:
Another technical detail of continuous variables relates to handling continuous random variables t... | $p_X(x)dx$ represents the probability measur $\mathbb{P}_X$ which is the probability distribution of the random variable $X$, it is defined by its action on measurable positive functions by
$$\mathbb{E}(f(X))=\int_{\Omega}f(X)d\mathbb{P}=\int_{\mathbb{R}}f(x)d\mathbb{P}_X(x)=\int_{\mathbb{R}}f(x)p_X(x)dx.$$
Now, we con... | {
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} |
Compute $\lim\limits_{n \to \infty} {\frac{1 \cdot 3 \cdot 5 \cdots(2n - 1)}{2 \cdot 4 \cdot 6 \cdots (2n)}}$ EDIT: @Holo has kindly pointed out that my concept of ln rules used in this question is wrong. However, the intuition behind using the tangent of a curve to find the sum to infinity of a series still stands. Th... | A completely different approach is to write
$${a_{n}} = \frac{1}{2} .\frac{3}{4} . \frac{5}{6} ...\frac{2n -1}{2n}=\frac {(2n)!}{(2^nn!)^2}$$
because you can divide a $2$ out of each term on the bottom and get $n!$ and you can multiply top and bottom by the bottom. Now feed it to Stirling
$$a_n\approx\frac{(2n)^{2n}e^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2907113",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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To be concave on $[0,1]$, $f(t)\leq f(0)+tf'(0)$ is enough? Suppose $f(t)>0$.
$f(t)\leq f(0)+tf'(0)$ if and only if $f$ is concave over $[0,1]$.
Is the above stetement true? There must be a counterexample in my
opinion.
| An obvious counterexample would be $f(t)=\sin(2\pi t)+1$, which is neither convex nor concave but fulfills your inequality.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2907200",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Relation between dilogarithm and its complex conjugate I am looking for a relation between the dilog and its complex conjugate, that is can I simplify the following summation of terms $$f(z) = \text{Li}_2(z) + (\text{Li}_2(z))^*?$$
I have looked through the many identities that are known to exist among such functions o... | The real part $\Re \mathrm{Li}_2(x) = \frac{1}{2}f(x)$ for $x>1$ can be computed with the Euler reflection formula (see https://en.wikipedia.org/wiki/Polylogarithm#Dilogarithm)
$$\Re \mathrm{Li}_2(x) = \frac{\pi^2}{6} - \mathrm{Li}_2(1-x) - \ln x \ln|1-x|$$
where I have used $\Re \ln(1-x) = \ln|1-x|$ for $x > 1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2907329",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Let us say that we are picking 2 letters from a set of 12. How would you describe the sample space? Let's say the first letter is from the following set: {A,B,C,D,E,F}. And the second letter is from the following set: {a,b,c,d,e,f}. Would this be simply $12\choose2$=66? Or would it be $_{12}P_2=132$? I tried to do it o... | Let $S = \{A, B, C, D, E, F\}$; let $T = \{a, b, c, d, e, f\}$. Then the sample space is
$$S \times T = \{(s, t) \mid s \in S, t \in T\}$$
that is, the set of ordered pairs in which the first element is a member of set $S$ and the second element is a member of set $T$. Since there are six choices for the first elem... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2907432",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Recognizing a $4\times4$ matrix Suppose I have
$$
\Sigma=\begin{pmatrix}\sigma_{11} & \sigma_{12} \\ \sigma_{21} & \sigma_{22}\end{pmatrix},\quad S=\begin{pmatrix}s_{11} & s_{12} \\ s_{21} & s_{22}\end{pmatrix}
$$
both of which are actually covariance matrices of two $2\times 1$ random vectors. So, in particular, $\Sig... | I might be wrong, but one way to do it may be through combining Kronecker product and Hadamard product:
$$ \biggl(\begin {bmatrix} 1 & 1 \end {bmatrix} \otimes S \otimes \begin{bmatrix} 1 \\ 1 \end {bmatrix}\biggr) \circ \biggl(\begin{bmatrix} 1 \\ 1 \end {bmatrix}\ \otimes \Sigma \otimes \begin {bmatrix} 1 & 1 \end {b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2907791",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that $\lim\limits_{s\to0^+}s^z=\lim\limits_{s\to 0^+} e^{z\ln s}=0$ where $z\in\mathbb C$ and $Re(z)>0$ Prove that $\displaystyle\lim_{s\to0^+}s^z=\lim_{s\to 0^+} e^{z\ln s}=0$ where $z\in\mathbb C$ and $Re(z)>0$
Using the $\epsilon-\delta$ definition we have
Let $\epsilon>0.$
We have to find $\delta>0$ such that... | Hint: What does the image of $\ln s$ for $s\in(0,\delta)$ look like in the complex plane? How does that vary with $\delta$.
What does the image of $z\ln s$ look like, given that $z$ has positive real part? How does that vary with $\delta$?
What is then the maximal size of $e^{z\ln s}$, given $\delta$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2907936",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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Given $\cos^2 x = 2 \sin x \cos x$, why can't I cancel $\cos x$ to get $\cos x = 2 \sin x$? If I have a function where I know $\cos^2 x = 2 \sin x \cos x$. Why can I not cross out $\cos x$ on both sides, because I get different values for $\cos x = 2 \sin x$?
| You may not divide the two members of an equation by $0$. So you can handle the problem with case analysis:
*
*if $\cos x=0$, the equation holds;
*else if $\cos x\ne0$, the equation can be reduced to $\cos x=2\sin x$.
Now you solve the two cases independently.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2908157",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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If we say "classes of non-zero integers modulo $n$", why does this not include the $0$ class? I suppose this is a bit of a wording question more than anything else - I'm working through group theory and was learning that "the (classes of) non-zero integers modulo $p$ form an Abelian group under multiplication." It's th... | I think the "blocking" isn't and shouldn't be "[non-zero integers] [modulo $p$]" but rather "[non-zero][integers modulo $p$]"
You have three [integers modulo $3$]. They are $[53], [-216]$ and $[3{,}691]$. $[53]$ and $[3{,}691]$ are non-zero [integers modulo $3$]. And $[-216]$ is not a non-zero [integer modulo $3$].
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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integral of $\int_{0}^{2014}{\frac{\sqrt{2014-x}}{\sqrt{x}+\sqrt{2014-x}}dx} $
Solve $\int_{0}^{2014}{\frac{\sqrt{2014-x}}{\sqrt{x}+\sqrt{2014-x}}dx} $.
Answer is 1007.
I tried multiplying $\sqrt{x}-\sqrt{2014-x}\;$,
which results in $\frac{\sqrt{2014-x}(\sqrt{x}-\sqrt{2014-x})}{2x-2014}=$$\frac{\sqrt{2014x-x^2}}{... | $$let \int_{0}^{2014}{\frac{\sqrt{2014-x}}{\sqrt{x}+\sqrt{2014-x}}dx}=A \\ put \ 2014-x=u \\ \implies \int_{0}^{2014}{\frac{\sqrt{2014-x}}{\sqrt{x}+\sqrt{2014-x}}dx}=\int_{2014}^{0}{\frac{\sqrt{u}}{\sqrt{u}+\sqrt{2014-u}}(-du)}\\=\int_{0}^{2014}{\frac{\sqrt{u}}{\sqrt{u}+\sqrt{2014-u}}(du)}=A \\ add \ both \ integrals \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2908398",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
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Monotonicity of the function $(1+x)^{\frac{1}{x}}\left(1+\frac{1}{x}\right)^x$. Let $f(x)=(1+x)^{\frac{1}{x}}\left(1+\frac{1}{x}\right)^x, 0<x\leq 1.$
Prove that $f$ is strictly increasing and $e<f(x)\leq 4.$
In order to study the Monotonicity of $f$, let
$$g(x)=\log f(x)=\frac{1}{x}\log (1+x)+x\log \left(1+\frac{1}{x... | Michael Rozenberg's answer has it all.
Here are two remaining proofs in Michael Rozenberg's answer:
*
*Show: $\ln{x}\leq\frac{2(x-1)}{1+x}$
We have $\ln(1+y)\leq\frac{2y}{{2+y}}$ for $-1<y<0$ (see e.g. in https://en.wikipedia.org/wiki/List_of_logarithmic_identities#Inequalities),
i.e. $\ln(x)\leq\frac{2(x-1)}{{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2908549",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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An Induction Problem, What Am I Supposed To Prove? I have encountered an induction problem which I don't understand. What I don't understand is what it is asking me to prove. I don't want a solution. The problem is:
If $u_1=5$ and $u_{n+1}=2u_n-3(-1)^n$, then $u_n=3(2^n)+(-1)^n$ for all positive integers.
Am I suppos... | You are supposed to prove $u_n=3(2^n)+(-1)^n$.
$u_1=5$ and $u_{n+1}=2u_n-3(-1)^n$ are the conditions you are supposed to make use of.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2908655",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Maximum value of $ab+bc+ca$ given that $a+2b+c=4$ Question:
Find the maximum value of $ab+bc+ca$ from the equation,
$$a+2b+c=4$$
My method:
I tried making quadratic equation(in $b$) and then putting discriminant greater than or equal to $0$. It doesn't help as it yields a value greater than the answer.
Thanks... | Without using calculus: Substituting $c=4-2b-a$, we get $$ab+bc+ca=ab+(a+b)(4-2b-a)=(4(a+b)-(a+b)^2)-b^2$$ and since $f(x)=4x-x^2=4-(x-2)^2$ has maximum at $(2,4)$, substituting $x:=a+b$ gives $$ab+bc+ca\le4-b^2\le4.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2908775",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 4
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$\int_{\mathbb R} \frac{1}{(1+ |t-y|)^r} \|f(y)\|_{L^1} dy \leq \frac{C}{r-1} \|f\|_{C(\mathbb R, L^1(\mathbb R))}$ for some $r>0$? Let $f:\mathbb R \to L^1(\mathbb R):t\mapsto f(t)\in L^1(\mathbb R).$ [So $f(t):\mathbb R \to \mathbb C: x\mapsto f(t)(x)$] Assume that $f$ is continuous with time variable $t$. Assume th... | By the norm on the continuous function mapping $\mathbb{R} \to L^1(\mathbb{R})$ I assume you mean the maximum-norm, i.e.
$$ \|f\|_{C(\mathbb{R},L^1(\mathbb{R}))} = \sup_{t\in \mathbb{R}} \|f(t)\|_{L^1(\mathbb{R})} $$
If that is the case, then the answer to your question is yes. Indeed, for any $t\in \mathbb{R}$ and $r ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2908892",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Inequality Involving $\int_V \left(\int_{B(x,\epsilon)} \eta_{\epsilon}(x-y) |f(y)|^p dy\right) dx $ I was reading the book "Partial Differential Equation" written by Lawrence C. Evans, coming up with a question.
On page 718, Evans wrote
$$\int_V \left(\int_{B(x,\epsilon)} \eta_{\epsilon}(x-y) |f(y)|^p dy\right) dx \\... | This is an application of Fubini's theorem, followed by the fact that the integrand is positive and $V$ is a subset of $W$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2909009",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Stuck On A Proof By Induction I need to prove true for all integers greater than and equal to 1 using induction.
I'll skip the base case, and the inductive assumption, and jump straight to the inductive step:
=
What I've done now is to say that is less than . But I don't know what to do from beyond there.
| One way by induction (as opposed to recognizing the inequality as just AM-GM for $1,2,\ldots,n\,$): write the inequality to prove as $\,\color{blue}{2^n n! \le (n+1)^n}\,$, and take this to be the inductive assumption. Then, to prove the inductive step for $\,n+1\,$:
$$
2^{n+1} (n+1)! = 2(n+1) \cdot \color{blue}{2^n ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2909233",
"timestamp": "2023-03-29T00:00:00",
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Let $f:[a,b]\to\Bbb{R}$ be continuous. Does $\max\{|f(x)|:a\leq x\leq b\}$ exist? Let $f:[a,b]\to\Bbb{R}$ be continuous. Does \begin{align}\max\{|f(x)|:a\leq x\leq b\} \end{align} exist?
MY WORK
I believe it does and I want to prove it.
Since $f:[a,b]\to\Bbb{R}$ is continuous, then $f$ is uniformly continuous. Let $\ep... | Because $[a,b]$ is compact, every sequence in $[a,b]$ has a subsequence that limits to a point in $[a,b]$. Pick a sequence $x_n \in [a,b]$ such that $\lim_{n \rightarrow \infty} |f(x_n)| = \sup_{[a,b]}|f|$. Now get a convergent subsequence $x_{n_k}$ that converges to some $x \in [a,b]$.
By continuity of $|f|$,
$|f(x)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2909330",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Find the probability $P(X
Suppose $X$ and $Y$ be two independent Poisson $(\lambda)$ random variables. Find $P(X<Y)$.
My attempt $:$
\begin{align*}P(X<Y) &= \sum\limits_{x=0}^{\infty} \sum\limits_{y=x+1}^{\infty} P(X=x,Y=y) \\
&= \sum\limits_{x=0}^{\infty} \sum\limits_{y=x+1}^{\infty} P(X=x) P(Y=y) \\
&= e^{-2\lamb... | For the special case of $X$ and $Y$ being identically distributed, you have
$$P(X < Y) + P(Y < X) + P(Y = X) = 1$$
$$2 P(X < Y) + P(X = Y) = 1$$
$$ P( X < Y) = 1/2 (1 - P(X = Y))$$
So it reduces to computing $P(X =Y)$ whose computation appears here Probability that two independent Poisson random variables with same par... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2909419",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Matrix eigenvalues Consider the matrix $$A_n=\begin{bmatrix}
a & b & 0 & 0 & 0 & \dots & 0 & 0 & 0 \\
c & a & b & 0 & 0 & \dots & 0 & 0 & 0 \\
0 & c & a & b & 0 & \dots & 0 & 0 & 0 \\
0 & 0 & c & a & b & \dots & 0 & 0 & 0 \\
0 & 0 & 0 & c & a & \dots & 0 & 0 & 0 \\
\vdots & \vdots & \vdots & \vdots & \vdots & \dd... | You got questions (a) and (b) already. For (c) the eigenvalues, you need the characteristic equation $\det (A_n - \lambda I) = 0$. This is the same as $D_n = \det (A_n) = 0$, if in there $a$ is replaced by $a-\lambda$. From your result,
$$
0 = D_n({\rm a \; replaced}) =\frac{1}{2^{n+1}\sqrt{(a-\lambda)^2-4bc}}[(a-\lam... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2909566",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the volume bounded above the sphere $r=2a\cos\theta$ and below the cone $\phi=\alpha$, where $0<\alpha<\frac{\pi}{2}$ Find the volume bounded above the sphere $r=2a\cos\theta$ and below the cone $\phi=\alpha$, where $0<\alpha<\frac{\pi}{2}.$
I'm supposed to use triple integrals in spherical coordinates to solve it... | HINT
I would use cylindical coordinates with
*
*$x=r\cos \theta$
*$y=r\sin \theta$
*$z=z$
and
*
*$0\le \theta \le 2\pi$
*$2a \cos^2 \alpha\le z\le 2a $
*$0\le r \le 2a \cos \alpha\sin \alpha$
*$dV=r\,dz\, dr\, d\theta$
that is
$$V=\int_0^{2\pi} d\theta\int^{2}_{2a \cos^2 \alpha}\, dz\int^{2a \cos \alpha\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2909665",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving that the sequence $\left \{ \frac{x_{n}}{y_{n}} \right \} \rightarrow \frac{x}{y}$ Suppose $\left \{ x_{n} \right \}$ and $\left \{ y_{n} \right \}$ converge to the limits $x$ and $y$, respectively. Also, suppose that $y_{n}$'s are nonzero. I want to show that the sequence $\left \{ \frac{x_{n}}{y_{n}} \right \... | You can proof
$$\lim_{n\to\infty}\frac{1}{y_n}=\frac{1}{y}$$
and then use the product rule.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2909775",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
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Proving that a function is continuously differentiable using decay of fourier series Let $\mathbb{S}^1=\mathbb{R}\backslash\mathbb{Z}$.
Let $\alpha$ be an irrational number, and consider the equation
$$g(x+\alpha)-g(x)=p(x), x\in \mathbb{S}^1$$
for an unknown function $g$, with a given function $p\in C^\infty(\mathbb{S... | Yo need to use the fact that $|e^{2 \pi i \alpha n} - 1|$ is comparable to the infimum over all natural numbers $m$ of
$$
|m - \alpha n | = n \, \Big| \frac{m}{n} - \alpha \Big|
$$
That quantity is measuring how googd can your number by approximated by rational numbers. You want to have some lower bound of the form:... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2909902",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How many roots does the equation $z^{2018}=2018^{2018}+i$ have?
Consider the equation
$$ z^{2018}=2018^{2018}+i$$ where $i=\sqrt{-1}$.
How many complex solutions as well as real solutions does this equation have?
My attempt:
I took the polar form as the equation has very difficult to handle when using $z=x+iy$.
S... | There are several red herrings in the question and, as the problem is stated, you don't need to describe the solutions.
Your problem can be generalized to $z^n=a+i$, where $n$ is a positive integer and $a$ is real. Clearly, $z$ cannot be real, nor can $a+i$ be zero.
Thus the solutions are all complex (not real) and the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2910047",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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How to apply limit properties in this case? I was asked to find the following limit:
$$
\lim_{x \to 1} \frac{1 - \sqrt x}{1 - x}
$$
I worked it out using direct substitution that the limit is $\frac{1}{2}$.
Initially I was trying a more algebraic approach, finding separately the limits of
$$
\lim_{x \to 1} 1 - \sqrt x... | Since the function involves a square root of $x$ we tacitly assume $x >0$ and from $x\to 1$ we assume that $x\neq 1$. Then we may use $1- x = (1 - \sqrt x)(1 + \sqrt x)$. It follows that
$$
\lim_{x \to 1} \frac{1 - \sqrt x}{1 - x} = \lim_{x \to 1} \frac{1 - \sqrt x}{(1 - \sqrt x)(1 + \sqrt x)} = \lim_{x \to 1} \frac{1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2910190",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Sum of all powers of two Prove that for any positive integer $n$, there exists a nonnegative integer $k$ with the property that $n$ can be written as a sum of the numbers $2^0,2^1,\dots,2^k$, each appearing once or twice.
It seems that we should begin with the canonical representation of $n$ as a sum of powers of two. ... | Basically, as you say, fill in the gaps. We can always write a positive integer $n$ as a sum of powers of $2$ using the binary expansion:
$$n = \delta_0 2^0 + \delta_1 2^1 + \ldots + \delta_k 2^k,$$
where $\delta_i \in \lbrace 0, 1\rbrace$. Take the least $m$ such that $\delta_i = 0$, and consider the least $l > m$ suc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2910274",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Suppose $r\lt1$. Prove that the series $\sum x_n$ is convergent. Let $x_n$ be a sequence in $\mathbb R$ and suppose $r = \lim_{n\rightarrow\infty} \root {n} \of {|x_n|}$ exists!
Suppose $r\lt1$. Prove that the series $\sum x_n$ is convergent.
I'm struggling to get started on this question, follow up questions include '... | Note, clearly $0\le r$. By $r \lt 1$, we know there exists $R \in (r,1)$ such that $0\le r \lt R \lt 1$. Now choose $\epsilon = R - r$ then by hypothesis, there exists an $N$ such that $\forall n \ge N$ we have:
$$|\root n\of{|x_n|} - r| \le R - r$$
Hence $|\root n\of{|x_n|}| \le R$ for all $n \ge N$ (Triangle Inequali... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2910413",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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If $x$ is less than $\pi/2$ then show that $i\cos^{-1}(\sin x + \cos x)$ has two real values Here's how I tried :
Let $$i\cos^{-1}(\sin x +\cos x) =y$$
So $$\cos^{-1}(\sin x + \cos x) = -iy$$
So $$\sin x +\cos x =\cos(iy)$$
Now
$$\sqrt{2} \cos\left(2n\pi + x- \frac{\pi}{4}\right)= \cos(iy)$$
What now?
| Notice that $$i\cos^{-1}(\sin x + \cos x)=i\cos^{-1}(\sqrt 2\cos (x-\dfrac{\pi}{4}))$$I assume it must also be that $x>0$ otherwise for example for $x=-\dfrac{\pi}{4}$ the expression would become zero and has one real value. If so, we have $$-\dfrac{\pi}{4}<x-\dfrac{\pi}{4}<\dfrac{\pi}{4}\\1<\sqrt 2\cos (x-\dfrac{\pi}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2910516",
"timestamp": "2023-03-29T00:00:00",
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"answer_id": 2
} |
About definition of Ergodic theorem Let $(X,\Sigma, \mu)$ be a probability space, and $T:X\rightarrow X$ be a measure-preserving transformation. We say $\mu$ is ergodic with respect to $T$ if
for every $E\in \Sigma $ with $T^{-1}(E) = E$, either $\mu(E)=1$ or $0$.
I have a very fundamental problem about this defin... | For any $E\in \Sigma$, let $P(E)$ be the implication "$\left(T^{-1}E=E\right)\Rightarrow \left(\mu(E)\in \{0,1\}\right)$". Ergodicity of $\mu$ with respect to $T$ means that fro all $E\in\Sigma$, proposition $P(E)$ is true. Since the first part of the implication involves $T$, the definition of ergodicity also implies ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2910648",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
General statements for the second derivative of a function I am working on a task about second derivative. The task is:
$f(x)$ on $(-1,1)$ has the values $f(-1)=-10$, $f(0)=-10$ and $f(1)=-3$.
What can you say about the values for first and second derivative?
For the first derivative I use the mean value theorem and f... | Hint
If we consider $x\in(-1,1)$ then
$$|f'(x)|\le k\implies |f(x)-f(0)|\le k\left|x\right|\le k\implies |f(1)-f(0)|\le k.$$
Since $f(1)-f(0)=7,$ what can we say about $k?$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2910796",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
characteristic function of a convolution of measures Take the probability measures $\mu,\nu$ on $\mathbb{R}$ and denote $\varphi_{\mu}$ (the same for $\nu$) its characteristic function. Why holds
$$\varphi_{\mu *\nu}(t)=\varphi_{\mu}(t)\cdot\varphi_{\nu}(t)$$
where $\mu*\nu$ denotes the convolution of $\mu$ and $\nu$?
| Neither do we need to assume that $\mu$ and $\nu$ admit densities with respect to a common reference measure, nor do we need to restrict this result to $\mathbb R$.
Remember that if $(E,\mathcal E)$ is a measurable group and $$\tau:E^2\to E\;,\;\;\;(x,y)\mapsto xy$$ denotes the group operation, then $\mu\ast\nu$ is the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2910885",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Groups of order $360$ have a subgroup of order $10$ I want to prove that groups of order $360$ must have a subgroup of order $10$.
By Sylow's theorem, the number of Sylow $5$-subgroups $n_5 \equiv 1 \pmod 5$ and $n_5\mid 360$. There are three solutions: $1, 6, 36$ (let me know if I missed any).
If $n_5=1$, then the onl... | If $n_5=6$ and $P$ is a Sylow 5-subgroup, you have $|N_G(P)|=60$. Set $H=N_G(P)$, and we find a subgroup of order $10$ in $H$. Your arguments work.
In $H$, $n_5$ is either $1$ or $6$. If $n_5=1$, multiply Sylow $5$-subgroup with a cyclic of order $2$. If $n_5=6$ and $Q$ is Sylow $5$-subgroup, then $|N_H(Q)|=10$. So, $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2911121",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
The limit of the function $ \frac{\log_{2}(x)-1}{x^2-4}$ as $x=2$ I want to solve the limit:
$$\lim_{x \to 2}{\frac{\log_{2}(x)-1}{x^2-4}} $$
The problem is I can't use L'Hospital and Taylor Series to solve it.
In My attempt I'm stuck here:
$$ \lim_{x \to 2}{\frac{\frac{\ln(x)}{\ln(2)}-1}{x^2-4}} $$
$$ \lim_{x \to 2}{\... | A very standard trick for calculating limits when L'Hopital or Taylor series are not allowed is using the definition of the derivative of some function at some point. Usually, this "some point" is the point at which you take the limit and the "some function" is the "complicated" one in the denominator.
In your case, i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2911227",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Number of solutions for a given logical equation I came across the following question while studying logic and cannot find a solution for it anywhere. I am studying by myself and think I just don't know exactly the right terms to search for it online (I'm not sure it is called a logical equation so excuse the title of ... | In addition to @Rushabh Mehta's answer:
note that $[P_i]$ can be seen as a "classic" (aka Pre-Calculus) function. For example, $f_1=[P_1]$ is simply the function $$f_1(x)=[P_1(x)]=\left\{ \begin{array}{rl} 1, & x \leqslant 5 \\ 0, &x>5 \end{array} \right.$$
Therefore, you can study the given equation $x=f_1(x)+2f_2(x)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2911315",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 3
} |
Doubt on the definition of ordered topology given in 'Foundations of Topology By C. Wayne Patty'
If the underlined symbol is as it is, The definition is confusing. Is it
$\mathscr T$ or $\mathscr S$? Please help me with the definition.
| All marked $\mathcal{T}$ should be $\mathcal{S}$, of course.
<rant>
It's too bad so many math-books are submitted as pdf's from Latex sources without the need for human copy editors any more. Huge savings, I know. But earlier all published maths texts went through human eyes during proof reading and type setting and s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2911407",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Finding value of $\lim_{n\rightarrow \infty}\frac{1+\frac{1}{2}+\frac{1}{3}+ \cdots +\frac{1}{n^3}}{\ln(n)}$
Find the value of
$$\lim_{n\rightarrow \infty}\frac{1+\frac{1}{2}+\frac{1}{3}+\cdots +\frac{1}{n^3}}{\ln(n)}$$
My Try: Using Stolz-Cesaro,
Let $\displaystyle a_{n} = 1+\frac{1}{2}+\frac{1}{3}+\cdots \cdots ... | To answer your question directly. The mistake is made in $a_{n+1}-a_n$. For example, if $n = 10$, then this is given by
$$a_{11}-a_{10} = 1/1331 + 1/1330 + \cdots + 1/1001$$
since $11^3 = 1331$ and $10^3=1000$.
If you insert this, you still are going to need to do something similar as Gabriel suggested by viewing the s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2911688",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 4
} |
Log transforming an ODE I'm doing some numerical simulations of an exponential growth like system which, for simplicity, has the form:
$$
\frac{dx}{dt}= ax + bxy \quad\quad \frac{dy}{dt}= cy + dxy
$$
For some parameter values i get instability in the simulation though I remember reading a paper which used log transfor... | (If you need more information, for example Lyapunov functions, this equation is similar to Lotka-Volterra equation.)
Dividing the first equality by $x$:
$$\frac{1}{x} \frac{dx}{dt}=a+by$$
i.e:
$$\frac{d \log(x)}{dt}=a+by$$
and similarly:
$$\frac{d \log(y)}{dt}=c+bx$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2912041",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Lower bound for complex polynomial beyond circle or radius R If we have a polynomial with $c_i$ a complex number
$$c_nz^n + c_{n-1}z^{n-1} + \cdots + c_1 z + c_0$$
then $$|P(z)| > \frac{|c_n|R^n}{2}$$
When |z| > R for some R
I have tried using the triangle inequality where I obtain,
$|P(Z)| \leq |c_n||z|^n + \cdots + |... | You have
$$\frac{|P(z)|}{|z|^n}
=\left|c_n+\frac{c_{n-1}}z+\frac{c_{n-2}}{z^2}+\cdots+\frac{c_0}{z^n}\right|.$$
Show that if $|z|$ is large enough, then
$$\left|\frac{c_{n-1}}z+\frac{c_{n-2}}{z^2}+\cdots+\frac{c_0}{z^n}\right|<\frac{|c_n|}2$$
etc.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2912165",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Confusion about Nelson's proof of Liouville's theorem Nelson's proof of Liouville's theorem (in the case $n=2$) is as follows:
Consider a bounded harmonic function on Euclidean space. Since
it is harmonic, its value at any point is its average over any sphere,
and hence over any ball, with the point as center. Giv... | Ok, let's calculate the volume $|A|$ of $A$. WLOG I let $z=0$ and $w=d > 0$. Then
$$
\frac{|A|}2 = \int_{d/2}^{r+d}\sqrt{r^2-(x-d)^2}\,dx - \int_{d/2}^{r}\sqrt{r^2-x^2}\,dx = \int_{-d/2}^{d/2}\sqrt{r^2-x^2}\,dx.
$$
Substituting $x = r\sin t$ gives
$$
\frac{|A|}2 = r^2\int_{-\arcsin(d/2r)}^{\arcsin(d/2r)}\cos^2t\,dt = \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2912258",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
To prove that $f(z) = constant$ if $f'(z) = 0$, why is it necessary to prove that $u$ and $v$ are constant for all paths? In Churchill's "Complex Variables and Applications", when proving the following statement:
$f'(z) = 0 \space\space \forall z \in \mathbb{D}\subset \mathbb{C} \implies f(z) = constant \space\space \f... | The problem here is that the C-R equations are only concerned with derivatives in the $x$ and $y$ directions but none of the infinitely many directions inbetween. Therefore it doesn‘t follow entirely trivially that $f$ is constant.
However, we can assume the set that $f$ is defined on to be open and locally path conne... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2912393",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
What does it mean to have an absolute value equal an absolute value? I have no problem reading absolute value equations such as $|x -2| = 2$.
I know this means that the distance of some real number is $2$ away from the origin. Because the origin splits the number line into a negative side and positive side then the nu... | Your interpretation is good.
Any value $v$ is at distance $|v|$ from origin. Sometimes we are given the distance and are asked to find original value. When something ($\in \mathbb{R}$) is at distance $|w|$ from origin, it has a value either $w$ or $-w$.
As you said, $|x-2|=2$ means that $(x-2)$ is at distance $2$ from... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2912507",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 0
} |
Show there is a bijection between $R^n\ and\ R^N$ Let $N=\{1,2,3,...,n\}$, show there is a bijection between $R^n\ and\ R^N$
Define $f: R^n \rightarrow R^N$ by $f(x_1,..,x_n)=g_{(x_1,...x_n)}$ where the n-tuple is an element of $R^n$ and $g_{(x_1,...x_n)}:N \rightarrow \mathbb{R}$ is defined by:
$g_{(x_1,...x_n)}(i)=x_... | I would just appeal to cardinal arithmetic. We have $$\mathfrak c = |\Bbb R|=2^{\aleph_0}=2^{|\Bbb N|}\\
\mathfrak c = |\Bbb R|\le |\Bbb R^n|\le |\Bbb R^{|\Bbb N|}|=(2^{|\Bbb N|})^{|\Bbb N|}=2^{|\Bbb N|{|\Bbb N|}}=2^{|\Bbb N|}=\mathfrak c$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2912564",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
An ant is to walk from $A$ to $B$. Calculate the number of paths. An ant has to walk from the left most lower corner to top most upper corner of $4 \times 4$ square. However, it can take a right step or an upward step, but not necessarily in order. Calculate the number of paths.
| Realise that the ant in question must take 4 steps to the right and 4 upwards. Representing a step towards right as R and an upward step as U, the ant can choose paths like RRRUUUUR, URURURUR, etc.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2912854",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Will this series converge? $\sum \frac {1/2 + (-1)^n}{n}$ Will this series converge?
$\sum \frac {1/2 + (-1)^n}{n}$
MY Try: Dirichilet , Abel , libnitz rules can not be used. $\sum \frac {1/2 + (-1)^n}{n} = \sum \frac {1}{2n} + \sum\frac {(-1)^n}{2n} $. Is it possible to write in that way? If yes the how?
Can any... | $$\sum\frac{1/2+(-1)^n}{n}=\frac{1}{2}\sum\frac{1}{n}+\sum\frac{(-1)^n}{n}$$
I believe that the first summation is called the harmonic series and is divergent whilst the second summation is convergent
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2913073",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Show that there is at most one entire function satisfying the following condition.
Show that there is at most one entire function $f:\mathbb C\to \mathbb C$ with $f(0)=2+3i$ satisfying $$f'(z)=\sin(z)f(z)+e^{z^2}$$ for all $z\in\mathbb C$.
My question: Although this is a problem I encountered in a course for complex ... | That's right, if you have a proof of Picard-Lindelöf that works for holomorphic functions in the plane.
Luckily you don't need any non-trivial theorems here, because first-order linear equations are trivial via integrating factors. Say $f_1$ and $f_2$ are two solutions, and $h=f_1-f_2$. Then $h'(z)-h(z)\sin(z)=0$, so i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2913192",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Iterations of a multivariable function How do you define iterations of multivariable functions?
To be clear(example):
If $f: \mathbb R^2 \to \mathbb R$
How do you define
$f \circ f$, or $f \circ \cdots \circ f$?
I admit that this question sounds very odd, but I think I need to define or learn of this. (Why? I want to... | The composition is undefined as $$f \circ f=\mathbb{R}^2\xrightarrow{f}\mathbb{R}\xrightarrow{?}\underline{?}.$$ However if you have a function $\mathbb{R}^2\xrightarrow{F}\mathbb{R}^2,$ then we can easily form the composition.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2913277",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find the radius of a cylinder of given volume V if its surface area is a minimum. this question is driving me crazy as I'm not sure how they've got the answer.
The surface area is given as $S = 2\pi r^2 + \frac {1}{50r} $ and they are asking for the value of r for which S is minimum.
The derivative of this (I hope!) is... | By the classical formulas, $$V=\pi r^2h$$ and $$S=2\pi r^2+2\pi rh.$$
Eliminating $h$,
$$S=2\pi r^2+\frac{2V}r.$$
Then cancelling the derivative,
$$4\pi r-\frac{2V}{r^2}=0$$ or $$r^3=\frac V{2\pi}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2913363",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Matrix notation $i$ $j$ Let $A = \begin{bmatrix}
a_1 & a_2 & \cdots & a_n \\
\end{bmatrix}$ be a $n \times n$ matrix such that $a_i \cdot a_i = 1$ for all $i$ and $a_i \cdot a_j = 0$ for all $i \neq j$.
I'm familiar with $i$ indicating row and $j$ indicating column but I'm not sure what these dot product... | Such $B$ doesn't fulfill your constraints. Perhaps you intended$$B=\begin{bmatrix}\dfrac{1}{\sqrt 2}&-\dfrac{1}{\sqrt 2}\\\dfrac{1}{\sqrt 2}&\dfrac{1}{\sqrt 2}\end{bmatrix}$$here we define $$a_m=\begin{bmatrix}b_{1m}\\b_{2m}\\.\\.\\.\\b_{nm}\end{bmatrix}\qquad,\qquad \forall m$$and $$a_i\cdot a_j=\sum_{k=1}^{n}b_{ki}b_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2913485",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
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