Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Plotting a line without negative values Is it possible to have a line a equation of sine without negative values. Not the same as sin(abs(x)) as negative values would be reflected in the x-axis. I'm looking for an equation where negative values of sin are converted to 0, looking not too dissimilar to a toblerone.
| What about $f(x)=\max(\sin(x),0)$?
Edit: The function given by @RideTheWavelet in his comment is precisely this one, stated in different form.
| {
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Finding a threshold between two accumulations My question:
I have a list of numbers. This numbers are part of two accumulations, for each accumulation there is some unknown number of values around a specific average I don't know.
How can I find a threshold between those two accumulations, so I can say for every number ... | I already have an idea:
Maybe I could "group" the numbers reducing their "resolution" and then calculate the threshold of the now bimodal distribution. But this "resolution" has to be right, if it's to small, the result would be too unprecise, if it's too high, the result could be totally wrong.
I'm interested in your ... | {
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Interpretation of Kolmogorovs Law of Large Numbers and Convergence My lecture slides begin,
Consider a sequence of random numbers $X_1,X_2,...,X_n$ (the lecturer here stated
that random numbers and random variables - which in my understanding
are maps from a sample space to the real line - can be used
interchangeably)... | I take it that the random variables $X_1,X_2,\ldots$ are assumed to be independent - this is an important assumption.
For a given element of the sample space, $\omega$, the sequence $X_1(\omega),X_2(\omega),\ldots$ is a sequence of real numbers. Saying that $X_n(\omega)\to X(\omega)$ is shorthand for saying that the li... | {
"language": "en",
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Can a topology $\tau$ on $\mathbb R$ be defined such that $(\mathbb R,\tau)$ is a compact Hausdorff space? Can a topology $\tau$ on $\mathbb R$ be defined such that $(\mathbb R,\tau)$ is compact Hausdorff ? Obviously such a topology must be Normal , but I am unable to make any further conclusion . Please help . Thanks ... | Max's answer is fine: If $f:X \rightarrow Y$ is a bijection and $Y$ has a compact Hausdorff topology $\mathcal{T}$ we can define $\mathcal{T}_X = \{ f^{-1}[O] : O \in \mathcal{T}\}$ as a topology on $X$. This makes $f$ a homeomorphism, and any properties that $Y$ has, $X$ has too in this topology.
But to give a concret... | {
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Find the volume of a revolved cycloid I have to find the volume of the area surrounded by the first arc of the cycloid given by $x(t) = a(t - \sin t), y(t) = a(1 - \cos t)$ when revolved around the $y$-axis. I know that:
$$ V = \pi \int\limits_{a}^{b}[f(x)]^2dx$$
The problem here is that I do not know which bounds I ne... | Your formula works only if a line parallel to the $x$-axis intersects the curve at a single point. But that is not the case here, because for every $y$ between $0$ and $a$ there are two intersections.
The integral
$$
V_1 = \pi \int_{0}^{2a}x_{int}^2 dy
= \pi \int_{0}^{\pi}a^3(t - \sin t)^2 \sin t\, dt
$$
gives the vol... | {
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Definition of hyperplane in machine learning On this answer the hyperplane, presumably in a perceptron classifier, is described as the dot product $\langle \vec{w_{x}},\vec{x} \rangle$, where $\vec{w_x}$ is presumably the vector of weights, and $\vec x$ an example in the training set.
My very tentative understanding is... | The hyperplane is defined by the equation $\langle \vec{w},\vec{x} \rangle = 0$. This hyperplane partitions the training set into two sets, $\{\vec x\mid \langle\vec{w},\vec{x}_i\rangle >= 0\}$, and $\{\vec x\mid \langle\vec{w},\vec{x}_i\rangle <0\}$.
| {
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eigenvalues of $Q+\mu cc^T$ as $\mu\rightarrow +\infty$ Assume $Q$ is a $n\times n$ SPD, $c$ is a $n\times 1$ matrix. Let $\mu>0$, then one of the eigenvalues of $Q+\mu cc^T$ will go to infinity as $\mu\rightarrow +\infty$ and the other eigenvalues will remain bounded.
Any hint on how to prove the claim?
| The characteristic equation of $Q+\mu cc^T$ is:
$$det((Q + \mu cc^T)-\lambda I)=det((Q-\lambda I) + \mu cc^T)=0.$$
It can be transformed, using matrix-determinant lemma (https://en.wikipedia.org/wiki/Matrix_determinant_lemma), into:
$$\left(1+\mu c^T(Q-\lambda I)^{-1}c\right)det(Q)=0$$
Because $det(Q)\neq0$, this is eq... | {
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Does domination of exponential-factorial by tetration generalize to higher-order hyperoperations? Let $\star$ be any operation in the sequence of hyperoperations $(\text{Succ},+,\times,\uparrow,\uparrow\uparrow,\ldots)$, and consider the $\star$-factorial function defined as follows on the positive integers:
$$\begin{a... | For convenience, we will let $a \uparrow^b 0 = 1$ and $a\uparrow^0 b = ab$.
As in the proof for the exponential case, we can prove that $3 \uparrow^{k+1} (n-2) < f_{\uparrow^k}(n) < 3 \uparrow^{k+1} (n-1)$ for $n \ge 2$. The left inequality is trivial, so we will prove the right hand side.
First, we need a lemma:
Lemm... | {
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Number of strings generatable I really do not know how to start with the conditions of this problem:
A H64 string is a string with 64 characters, each character must be
choosen from 16 hexadecimal characters (0 - 9, a - f). However, no
character can appear over 10 times. How many H64 strings that can be
generate... | This is not a full answer. But I think there are enough hints to work it out.
Let $A_k$ denote the set of such valid H64 strings that make use of $k$ distinct hexdigits. By pigeonhole principle, the given condition leads to the constraint that $k\ge6$. For $k\neq l$ clearly, the two sets of strings $A_k$ and $A_l$ are... | {
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How can I find a closed form for this sum $\sum_{k=1}^n{n \choose k}p^{k+1}(1-p)^{n-1}$? Based on some guess and check I think that
$$\sum_{k=1}^n{n \choose k}p^{k+1}(1-p)^{n-1}=p(1-p)^{n-1}((p+1)^n-1)$$
Where $0\leq p \leq 1$
but I'm not sure how to get from one to the other, or if it is truly correct.
| By the Binomial Theorem, we have that
$$\begin{align}
\sum_{k=1}^{n}\binom{n}{k}p^{k+1}(1-p)^{n-1}&=p(1-p)^{n-1}\sum_{k=1}^{n}\binom{n}{k}p^{k}\\
&=p(1-p)^{n-1}\sum_{k=1}^{n}\binom{n}{k}p^{k}(1)^{n-k}\\
&=p(1-p)^{n-1}\left[\sum_{k=0}^{n}\binom{n}{k}p^{k}(1)^{n-k}-1\right] \quad \text{ since }\binom{n}{0}p^{0}(1)^{n-0}=... | {
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Why does $f$ strictly increasing implies the triangular inequality for this metric? Assume $(X,d)$ is a metric space, and define a new metric $\tilde{d}$ on $X$.
Set $\tilde{d} = \frac{d(x,y)}{1+d(x,y)}$. Now with manipulation and since $d$ is a metric, I manage to show that $\tilde{d}$ satiesfies the triangular ineqal... | Let $u=d(x,z),v=u=d(x,y),w=d(y,z)$. It is enough to check the condition $$u\le v+w\implies f(u)\le f(v)+f(w).$$ By increasingness we have $f(u)\le f(v+w)$. Elementary computation shows that $f(v+w)\le f(v)+f(w)$, which finishes this argument.
| {
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How do you quickly know that this matrix is diagonalisable (characteristic polynomial given)? We have matrix $A=\begin{pmatrix}
3 & 4 & -3\\
2 & 7 & -4\\
3 & 9 & -5
\end{pmatrix}$
The characteristic polynomial is $-(\lambda-2)^{2} \cdot (\lambda-1)=0$
Now I'd like to know a quick way to know if this matrix is diagona... | I think that there's a simple test that requires really little further computation. If you want to check if $\lambda$ is a double eigenvalue of $A$, then it is sufficient to see if $A-\lambda I$ has rank $n-2$.
In your case, if $A-2I$ has rank 1, and rank 1 matrices are really simple, since all the columns are multiple... | {
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WolframAlpha says limit exists when it doesn't? I was trying to calculate the following limit:
$$
\lim_{(x,y)\to (0,0)} \frac{(x^2+y^2)^2}{x^2+y^4}
$$
and, feeding it into WolframAlpha, I obtain the following answer, stating the limit is $0$:
However, when I try to calculate the limit when $x = 0$ and $y$ approaches... | This is only to complement the excellent answer of StackTD, who correctly shows that you are right — the limit does not exist (as one can find two different paths to the origin along which the limits of the function differ). The key message is:
Do not try limits in more than one variable with Mathematica or WolframAlp... | {
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Reparametrization of a curve Given a parametrized curve, we know that its arc length parametrization is its unit speed reparametrization. However, I wanted to know if there was any generic procedure to find any other reparametrizations of the same curve, which are not unit speed, and are non trivial?
| Assume you have a curve $\gamma : [a,b] \to \mathbb R^d$ and $\varphi : [a,b] \to [a,b]$ is a reparametrization, i.e., $\varphi'(t) > 0$. Then you can prescribe any speed function for your parametrization. Given a function $\sigma: [a,b] \to \mathbb R_{>0}$, define $\varphi$ via the ODE
$$ \varphi'(t) = \frac{\sigma(t)... | {
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Using Lagrange Multipliers to determine the point on a surface nearest to P I'm attempting to figure this problem out. I would appreciate some guidance on how to get the answer. Thanks.
Consider the surface defined as $S: x^2+y^2+z^2 = 8$. If we have a $P = (0,1,1)$, use Lagrange multipliers to determine the point on $... | It is obvious that the shortest distance between $P$ and $S$ is the distance between $P$ and the center of the sphere, minus the radius, that is:
$$
|\sqrt{0^2+1^2+1^2}-\sqrt{8}| = \sqrt{2}
$$
Now, if you want the coordinates of the nearest point, you can use Lagrange multipliers indeed: you want to minimize the (squ... | {
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Is my understanding of weights and weight spaces of subalgebras of $\mathfrak{gl}(V)$ correct? I am learning about weights of subalgefbras of $\mathfrak{gl}(V)$ where $V$ is a vector space over some field $\mathbb{F}$.
The definition that I have read states
Let $M$ be a subalgebra of $\mathfrak{gl}(V)$. A weight of $M$... | Yes, that is exactly correct. Each $V_\lambda$ is a simultaneous eigenspace for every linear map in $M$. $\lambda$ is the function that assigns to every linear map in $M$ the eigenvalue associated to its action on vectors in $V_\lambda$.
This example may be helpful: Take $V = \mathbb C^3$ and take
$$M = \left\{ \left( ... | {
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Derivative of $x|x|$ I am trying to find the derivative of $f(x)=x|x|$ using the defition of derivative. For $x > 0$ I found that $f'(x)=2x$ and for $x<0$ the derivative is $f'(x)=-2x$. Everything is fine up to here. Now I want to check what happens when at $x=0$.
By the way, I know that $|x|$ is not differentiable at ... | You didn't do anything wrong, you in fact shown that $f$ is differentiable at $x = 0$ and $f'(0) = 0$. The fact that $x \mapsto |x|$ is not differentiable at $x = 0$ doesn't mean that if you consider the product of this function with another function then the result won't be differentiable at $x = 0$, as you have just ... | {
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How can I prove $\lim_{n \to \infty} \frac{n^2}{2^n}=0$ How can I prove $\lim_{n \to \infty} \frac{n^2}{2^n}=0$
I tried to use $\mid \frac{n^2}{2^n} - 0 \mid <\epsilon $ However, because of $n^2$ I cannot use it. Also, I tried to use ratio test and I got $\lim_{n->\infty}\frac{(n+1)^2}{2}\frac{1}{n^2}$, but after that ... |
Stolz-Ces$\mathrm{\grave{a}}$ro Theorem:
$$
\lim_{n \to \infty}{n \over 2^{n}} =
\lim_{n \to \infty}{\left(n + 1\right) - n \over 2^{n + 1} - 2^{n}} =
\lim_{n \to \infty}{1 \over 2^{n}} = \bbox[#ffe,10px,border:1px dotted navy]{\displaystyle{\large 0}}
$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Solve $\int_{0}^{\infty}\frac{\ln(2x)}{4+x^2}dx$ by contour integration I'm a little stuck with this one. I've found the singularities to be at $\pm 2i$ and $0$ (branch point). So far, using a branch cut at $2\pi$ I've found that
$$\int_{0}^{\infty}\frac{\ln(2r)}{4+r^2}dr+\int_{0}^{\infty}\frac{\ln(2r)+2\pi i}{4+r^2}dr... | Consider
$$f(z) = \frac{\log(2z)}{z^2+4}$$
By using a key-hole integral with branch-cut on positive axis we should get
$$\int^\infty_0 \frac{\log(|2x|)}{x^2+4}\,dx + \int_{\infty}^{0} \frac{\log(|2x|)+2\pi i}{x^2+4}\,dx =2\pi i \sum \mathrm{Res}(f,z_0)$$
We see that the first and second integrals will cancel. Now to ... | {
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Which of the following sets of functions from $\mathbb{R}$ to $\mathbb{R}$ is a vector subspace of $\mathbb{R}$?
Which of the following sets of functions from $\mathbb{R}$ to $\mathbb{R}$ is a vector subspace of $\mathbb{R}$?
$S_1=\{f|\displaystyle\lim_{x\to 3}f(x)=0\}$, $S_2=\{h|\displaystyle\lim_{x\to 3}h(x)=1\}$, $... | Your solution is correct.
For $S_3$ : Take $g,h \in S_3$.
Then $\lim_{x \to 3} g(x)$ and $\lim_{x \to 3} h(x)$ both exist.
Now check
*
*$\lim_{x \to 3} (g(x)+h(x))=\lim_{x \to 3} g(x)+\lim_{x \to 3} h(x)=\text{exists}.$ This implies that $g+h \in S_3$.
*let $a \in \Bbb R$ be arbitrary. Then $\lim_{x \to 3} ag(x)=a\... | {
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How to prove that $f_n(x):=x^n$ is not a Cauchy sequence in $C[0,1]$ under the norm $\|f\|= \sup|f(x)|$?
How to prove that $f_n(x)=x^n$ is not a Cauchy sequence in $C[0,1]$ under the norm $\|f\|= \sup_{x\in [a,b]}|f(x)|$, by showing that it does not satisfy the definition of a Cauchy sequence?
| For a fixed $n \ge 1$ and for $m \ge n$,
$$
\|x^n-x^m\|=\max_{x\in[0,1]}(x^n-x^m) \ge (x^n-x^m)|_{x=1/2^{1/n}}
= \frac{1}{2}-\left(\frac{1}{2}\right)^{m/n},
$$
Hence,
$$
\|x^n-x^m\| \ge \frac{1}{4} \mbox{ whenever } \left(\frac{1}{2}\right)^{m/n} \le \frac{1}{4},
$$
which occurs whenever ... | {
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Solution of a trigonometric equation involving double sines,cosines What is the sum of all the solutions of $$\sin^2 (2\sin (x-\frac {\pi}{6}))+\sec^2 (x-\frac {\pi}{2}\tan^2 (x))=1$$ in $[0-4\pi]$ writing $\sec (..)=\frac {1}{\cos (..)} $ .. and rearranging we have $\cos (a).\cos (b)=1$ now as $\cos $ is $-1\leq \cos^... | We have $$\sin^2\left(2\sin\left(x-\dfrac\pi6\right)\right)+\tan^2\left(x-\dfrac\pi2\tan^2x\right)=0$$
For real $x,$
$\sin\left(2\sin\left(x-\dfrac\pi6\right)\right)=\tan\left(x-\dfrac\pi2\tan^2x\right)=0$
$\implies 2\sin\left(x-\dfrac\pi6\right)=m\pi\ \ \ \ (1)$ where $m$ is any integer
Now as $-1\le\sin y\le1$
$-1\le... | {
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Counting subsets of a given set Could anyone be kind in helping me with this simple question?
What is the number of subsets of $$\{1,2,3,\cdots, n\}$$ containing k elements?
Thanks.
| It is the sum of number of ways of choosing $r$ elements from n elements, where $r$ ranges from $0$ to $n$. This is because $r = 0$ is the null set, $k = 1$ is all $nC1$ subsets of cardinality $1$, and so on...
Just imagine yourself picking all subsets of size 0, then size 1, ..., till size n.
This is also the sum of t... | {
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Help me understand linear function. today in my high school lesson I learned for linear function. I know that each linear function has the form of $f(x)=kx+n$ where $k,x,n ∈ \mathbb R$. Now let's say be have those two functions:
$f_1(x)=kx+n_1\\
f_2(x)=kx+ n_2$
Since $k$ is same in both functions, we know that it repre... | I always make a drawing if possible and it is possible in this case, since you are working in the plane. Drawing two parallel lines, what you see is that the vertical distance is always constant (I included an example of such a drawing).
The vertical lines correspond with points on the $x$-axis, so let us take such a ... | {
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"timestamp": "2023-03-29T00:00:00",
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A hard integral Looking for a solution for an integral:
$$I(k)=\int_0^{\infty } \frac{e^{-\frac{(\log (u)-k)^2}{2 s^2}}}{\sqrt{2 \pi } s \left(1+u\right)} \, du .$$
So far I tried substitutions and by parts to no avail.
| Here is a start: $I(0) = \frac{1}{2}$
Proof:
$$I(0) = \int\limits_0^\infty \frac{\exp\left[-\frac{(\log u)^2}{2s^2}\right]}{\sqrt{2\pi} s (1+u)} \rm{d}u$$
Put $\log u = x$
\begin{align}
I(0) &= \int\limits_{-\infty}^\infty \frac{\exp\left[-\frac{x^2}{2s^2}\right]}{\sqrt{2\pi} s} \frac{e^x}{1+e^x} \rm{d}x \\
&= \int\lim... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2180737",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Combinatorics tennis match The prompt says, a tennis club has to select 2 mixed double pairs from a group of 5 men and 4 women. In how many ways can this be done?
There's total of 9 people and we need to choose of 8 people, that's what I think "2 mixed double pairs" means since one pair is 2 people and 2 double pairs w... | The amount of ways to pick the first pair is simply 5*4=20.
The next one is simply 4*3=12. (You have to chose one man and one woman for each pair.)
But we've overcounted by two (the same pair of pairs situations), so it's $\frac{20\cdot 12}{2}=\boxed{120}.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2180850",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Direct limit isomorphism Suppose I have a directed system $(V_i, \phi_i: V_i \rightarrow V_{i+1})$, say of vector spaces $V_i$. Let $\psi_i: V_i \rightarrow V_i$ be isomorphisms. I can construct the related directed system
$(V_i, \phi_i\circ \psi_i: V_i \rightarrow V_{i+1})$. Is it true that the direct limits $\lim_{\... | Take $V_i=k^2$ for all $i$, with $\phi_i$ given by the matrix $\begin{pmatrix}1&0\\0&0\end{pmatrix}$ for all $i$. Then $\varinjlim_{\phi_i}V_i$ is one dimensional.
But now take $\psi_i$ to be the isomorphism given by the matrix $\begin{pmatrix}0&1\\1&0\end{pmatrix}$ for all $i$. Then $\varinjlim_{\phi_i\circ\psi_i}V_i$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2180965",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
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Fibonacci Numbers proof, circular reasoning? I got this from "Number Theory" by George E. Andrews (1971), at the end of the first chapter, he asks for proofs by mathematical induction about Fibonacci Numbers as exercices. In one of them I am asked to show that
$$(\forall \, n \in \Bbb Z^+)((F_{n+1})^2-F_nF_{n+2}=(-1)^... | As @ElliotG pointed out, you cannot begin with what you want to prove. Instead, if you want to show an equality, begin with the left hand side of the inequality, then do things with it until you arrive at the right hand side.
In the induction step you know that $F_{n+2} = F_{n+1} + F_n$ and $F_{k+1}^2-F_k F_{k+2} = (-1... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Are all finite dimensional algebras over the real numbers `Banach algebra'-able Suppose that $A$ is a finite dimensional algebra over the real or complex numbers. Then $A$ has a natural topology induced from it being a finite dimensional vector space. Is it always true that there is a norm on $A$ satisfying $\| MN \| \... | Since $A$ is a finite-dimensional vector space, we can define a norm (any $\ell^p$-norm will do). Since $A$ is finite-dimensional, any linear operator is continuous with respect to this norm. To each $a\in A$ we can associate a linear map $x\mapsto ax$. Thus considering $A$ as a subalgebra of $L(A)$ with the operator... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Degree of $\mathbb{Q}(\xi_{p^{2}})$ over $\mathbb{Q}$. What is the degree of the extension $\mathbb{Q}(\xi_{p^{2}})$ over $\mathbb{Q}$ where $p$ is a prime and $\xi_{p^{2}}$ is a primitive $p^{th}$ root of unity?
| The degree of $\zeta_n$ over $\mathbb{Q}$ is $\phi(n)$ (Euler-phi function), for $n=p^2$ that is $p(p-1)$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Polynomial is irreducible over $\mathbb{Q}$ I've been scratching my head lately over this problem in my textbook:
Prove that the polynomial $x^4 + x^3 +x^2 +x +1$ is irreducible over $\mathbb{Q}$.
I've done some research and found this link, but they talk about Eisenstein's criterion, which we haven't covered in our ... | Note that in this case you can use Cohn's irreducibility criterion for base $b=2$, because $f(2)=31$ is a prime and the irreducibility follows.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2181384",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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The set of Polynomials is not complete in sup-norm over [0,1] Let $P$ be the set of all polynomials and consider the sup-norm on $[0,1]$ by $$||p||_\infty = \sup\{|p(x)| : x \in [0,1]\}.$$ I need to show $P$ is not complete in the sup-norm on the interval $[0,1]$. To do this, it suffices to show there exists a sequence... | Indeed you are trying to show that set of all polynomials complete! you have to show that the limit of a series of polynomial is not a polynomial. One of the ways is proving by counterexample. We have
$$\lim_{n\rightarrow \infty} 1-x^2/2+x^4/4-\cdots=cosx$$ which is not polynomial then the space is not complete.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2181493",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How many barcodes are there? A barcode is made of white and black lines. A barcode always begins and ends width a black line. Each line has is of thickness 1 or 2, and the whole barcode is of thickness 12.
How many different barcodes are there (we read a barcode from left to right).
I know that I'm required to show so... | Denote by $b_i$ the number of bars (black or white) of width $i\in\{1,2\}$. Then $b_1+2b_2=12$, hence $b_1$ is even. Since the total number of bars $b_1+b_2$ is odd it follows that $b_2$ is odd. This leaves the cases
$$(b_1,b_2)\in\bigl\{(10,1),(6,3),(2,5)\bigr\}\ .$$
The total number of admissible arrangements then co... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why is a number that is divisible by $2$ and $3$, also divisible by $6$? Why is it that a number such as $108$, that is divisible by $2$ and $3$, is also divisible by $6$?
Is this true that all numbers divisible by two integers are divisible by their product?
| Let $a$ be divisible by $2$ and $3$. So $$a=2^{r_{0}}\cdot p^{r_{1}}_{1}\cdot p^{r_{2}}_{2}\cdot\ldots=3^{s_{0}}\cdot q^{s_{1}}_{1}\cdot q^{s_{2}}_{2}\cdot\ldots$$ where $p_{i}$ and $q_{j}$ are distinct prime numbers, $r_{i}$ and $s_{j}$ are possibly $0$ for $1\leqslant i,j<\infty$ and $r_{0},s_{0}\geqslant 1$. We know... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Substitution or comparison? For the question,
$X^4 + \frac{9}{X^4} = (X^2 - \frac{a}{X^2})^2 + b$ I have to find the values of $a$ and $b$. I tried two solutions both included the expansion of brackets.
By comparing terms I obtained $3$ for $a$ and $6$ for $b$.
However when I tried to substitute using $b$, I obtained ... | $X^4 + \frac{9}{X^4} = (X^2 - \frac{a}{X^2})^2 + b$
$x^4 + \frac {9}{x^4} = x^4 + \frac {a^2}{x^4} + b - 2a$
So by comparing $a^2 = 9$ and $b-2a = 0$.
So $a = 3$ or $a = -3$ and $b = 6$ or $b = -6$.
I'm not sure what you mean by substituting? Do you mean picking an arbitrary value for $x$ and getting two equations fo... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Summation related to arctan What is $$\sum _{r=1} ^{90} \arctan \left(\frac{2r}{2+r^2+r^4}\right)$$ . Now I tried to express it as sum of $\arctan(a)-\arctan(b)$ . My try was using $r+\frac{1}{r}=a,r-\frac{1}{r}=b$ everything worked well except the term independent of variable. .all the problem is around that 2 in deno... | Hint:
$$\frac{2r}{2+r^2+r^4} = \frac{(r^2+r+1)-(r^2-r+1)}{1+(r^2+r+1)(r^2-r+1)}$$
Now you may proceed!
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solving differential equation: $y''-\frac{\epsilon ^2 y}{x^2} + \frac{y'}{x}=0$ I've derived a differential equation, as below.
$$y''(x)-\frac{\epsilon ^2 y(x)}{x^2} + \frac{y'(x)}{x}=0$$
It is a second order linear differential equation, WolframAlpha seem to suggest that the solution is (for the equation without $\ep... | This is a Cauchy-Euler equation.
Putting it in that form:
$$x^2\frac{d^2 y}{dx^2}+x\frac{dy}{dx}-\epsilon^2 y=0$$
You can use the ansatz $y=x^{\lambda}$:
$$\frac{dy}{dx}=\lambda x^{\lambda-1}$$
$$\frac{d^2y}{dx^2}=\lambda(\lambda-1)x^{\lambda-2}$$
To obtain the polynomial:
$$-\epsilon^2+\lambda^2=0$$
Solving for $\lamb... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Chinese Remainder Theorem Puzzle Im working on another CRT problem and I'm having a bit of trouble understanding the question at hand.
a group of seven men have found a stash of silver coins and are trying to share the coins equally among each other. Finally, there are six coins left over, and in a ensuing fight, one m... | You have interpreted three of the conditions properly, but you missed the original fight when there were seven men and six coins left, so you should add in $x \equiv 6 \pmod 7$ to your system.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proximal Operator / Mapping of $\frac{1}{2} {\|x\|}^2 + \delta_{\mathbb{R}_+^n}\left(x\right)$: Sum of $L_2$ Norm Squared and Indicator Function Let $$f(x) = \frac{1}{2}\|x\|^2 + \delta_{\mathbb{R}_+^n}(x)$$
(componentwise nonnegtive). How to find $$\operatorname{prox}_{\alpha, f}(z)$$
I know
*
*$\operatorname... | For simplicity I'll assume that $\alpha = 1$. You want to evaluate
$$
x^\star = \arg \min_x \quad I(x) + \frac12 \|x\|^2 + \frac12 \|x - z \|^2
$$
where $I$ is the indicator function of the nonnegative orthant.
We'll combine the two quadratic terms into a single quadratic term by completing the square. Notice that
\be... | {
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Stronger version Goursat Theorem Show the following:
Let $\Delta \subset \mathbb C$, be a triangle and let $f:\Delta \rightarrow \mathbb C$ be continous. Futhermore, assume that $f$ is holomorphic in the interior of $\Delta$. Then
$$\int _{\partial \Delta}f = 0$$
My attempt: By Goursat Theorem the result is valid for a... | I would probably complete your argument like this:
Since $f$ is continuous on $\Delta$, and $\Delta$ is compact, we know that $f$ is uniformly continuous on $\Delta$. So for any given $\epsilon > 0$, there exists a $\delta > 0$ such that
$$ |x - x_0 | < \delta \implies |f(x) - f(x_0)| < \epsilon $$
for all $x , x_o \in... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proof of the Kellogg's theorem Kellogg' theorem: Let $m\ge1,0<\alpha<1$. If $\Omega\in \mathbb{R}^n$ is of class $C^{m,\alpha}$, then $\log k\in C^{m-1,\alpha}(\partial\Omega)$, where $k(y)=k_{x_0}(y)=k(x_0,y)$ is the Poisson kernel for the domain $\Omega$ and $x_0\in\Omega$ is fixed.
I am looking for the proof of this... | Kellog's original article on his theorem.
http://www.ams.org/journals/tran/1931-033-02/S0002-9947-1931-1501602-2/home.html
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Optimal strategy for meta paper-scissor-rock You'll be familiar with the childhood game of paper-scissor-rock (hereafter abbreviated to PSR).
Let an 'optimal strategy' for PSR be one which will not be expected to lose in the long term against any other strategy.
It seems intuitively obvious that the optimal strategy fo... | This is a zero-sum game with payoff matrix $$\pmatrix{0&1&1&1&-1\\
-1&0&-1&1&1\\
-1&1&0&-1&1\\
-1&-1&1&0&1\\
1&-1&-1&-1&0}$$
where the rows are diamond, rock, paper, scissors, charcoal.
There are standard techniques to solve such games (easily Googleable!); a Nash equilibrium here is to go diamond or charcoal with $p=\... | {
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Proving the inequality $\int_{1}^{b} a^{\log _bx}dx$ > ${\ln b}$ Proving the inequality
$$\int_{1}^{b}a^{\log _bx}dx > {\ln b} : a,b>0 , b\neq1$$
Solving the integral, I found the result below,
$$\frac{\ ab-1}{{\ln ab}}{\ln b}$$
I know that I need only prove that, this part $$\frac{\ ab-1}{{\ln ab}}$$ must be greater ... | I leave this as an answer, since it is the only way for me to include an image. I think the problem in the question is invalid, and should be updated to be correct (or removed if the true problem in itself is as stated).
I let Mathematica plot the region where
$$
\ln b\frac{ab-1}{\ln(ab)}<\ln b.
$$
It is the blue domai... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How do we find the inverse Laplace transform of $\frac{1}{(s^2+a^2)^2}$? How do we find the inverse Laplace transform of $\frac{1}{(s^2+a^2)^2}$? Do I need to use the convolution theory? It doesn't match any of the known laplace inverse transforms. It matches with the Laplace transform of $\sin(at)$ but I don't know if... | $\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newc... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is there a list of common identities for principal branches of complex logarithms and roots? Often I have a necessity to simplify some expressions involving roots or logarithms, where arguments appear to be imaginary or even more generally complex. Due to $n$th root and logarithm being generally multivalued, many simpl... | There can't really be any such identities written down in a useful way.
The common ways of writing down things results in expressions that define analytic functions, so if we have an identity $f(z)=g(z)$, the expression $f(z)-g(z)$ will define an analytic function that is identically zero in the domain where the identi... | {
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How many six-digit numbers can I create using these digits {$1$, $4$, $4$, $5$, $5$, $5$, $7$, $9$} Step 1 - Determine how many six-digit numbers can we create if we had $8$ distinct digits
|{$1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$}| = 8
n!/(n-p)!
8!/(8-6)! = $20160$
BUT I have {$1$, $4$, $4$, $5$, $5$, $5$, $7$, $9$... | Case $1$: pick six digits from the $3$ distinct digits and $(5,5,5)$
$\dfrac{6!}{3!}$
Case $2$: pick six digits from the $3$ distinct digits and $(5,5,4)$
$\dfrac{6!}{2!}$
Case $3$: pick six digits from the $3$ distinct digits and $(5,4,4)$
$\dfrac{6!}{2!}$
Case $4$: pick six digits from $2$ distinct digits and $(5,... | {
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"timestamp": "2023-03-29T00:00:00",
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find the sum to $n$ terms of the series $1+4w+9w^2+...+n^2w^{n-1}$ where $w$ is $n$th root of unity I want to find the sum to $n$ terms of the series
$$1+4w+9w^2+...+n^2w^{n-1}$$
where$w$ is $n$th root of unity.
Let $$S_n = 1+4w+9w^2+...+n^2w^{n-1}$$
then $$ wS_n=w+4w^2+....+(n-1)^2w^{n-1}+n^2$$
therefore $$(1-w)S_n=1+... | I find $(1-w)S_n=-n^2+1+3w+5w^2+...+(2n-1)w^{n-1}$
Let $U_n=1+3w+5w^2+...+(2n-1)w^{n-1}$
$(1-w)U_n=1+2(w+w^2++w^{n-1})-(2n-1)=-2n+2(1+w+w^2++w^{n-1})$
Now $1+w+w^2++w^{n-1}=\dfrac{1-w^n}{1-w}=0$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Localization of a Dedekind Domain I have a question about this article here:
In the proof of (ii) of corollary 5.3 it says that $R/P^r\cong R_P/R_P P^r$ where $R_P$ is the localization of a Dedekind Domain R at the prime ideal P. Can someone explain this to me?
Thanks.
Edit: And I also don't really understand why the q... | Let $\phi: R \rightarrow R_P /R_P P^r, \ x \mapsto x + R_P P^r$ and $\mu: R \rightarrow R_P, \ x \mapsto \frac{x}{1}$. Then we have
$$ ker(\phi) = \mu^{-1}(R_P P^r) = P^r. $$
By the first isomorphism theorem for rings you get $R/P^r \cong R_P /R_P P^r$ as rings.
Edit:
Let $I\subseteq R_P/R_P P^r$ be an ideal and $\psi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2183355",
"timestamp": "2023-03-29T00:00:00",
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Is there a particularly good function form for this curve? This curve shape seems to appear in various natural phenomena:
Do you recognize it? Do you know a specific function form that could match it or approximate it closely?
| FIRST PART : Search for a model function.
In the empirical approach, the curve given by L_R_T is used (without the scattered points) : copy on Figure 1 below, curve drawn in red.
On Figure 2, instead of $x$ the abscissas are $\ln(x)$. The curve tends to become sinusoidal. But it is not symmetrical relatively to the hor... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2183483",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If $f$ is smooth and bijective, is $f^{-1}$ smooth, too? Let $U \subseteq \mathbb{R}^m$ be open. Assume that the function $f \colon U \rightarrow f(U)$ is $\mathcal{C}^{\infty}$ and a bijection and its differential $\text{d}f(x)$ is injective for every $x \in U$. Furthermore, assume that $f$ and $f^{-1}$ are continuous... | Write $f^{-1}=:g$. The inverse function theorem contains the formula
$$dg=\iota\circ df\circ g\ ,\tag{1}$$
whereby $\iota$ denotes taking the inverse of a regular linear map $L:\>{\mathbb R}^n\to{\mathbb R}^n$. As $f$ and $\iota$ are $C^\infty$ formula $(1)$ can be used to set up an induction proof that $g$ is $C^\inft... | {
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"timestamp": "2023-03-29T00:00:00",
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Number of 5-card Charlie hands in blackjack A five-card Charlie in blackjack is when you have a total of 5 cards and you do not exceed a point total of 21. How many such hands are there? Of course, the natural next question concerns six-card Charlies, etc.
It seems like one way of determining the answer might be to de... | Here is an answer by brute force enumeration of all 5-card hands, using an R program: the number of 5-card Charlie hands is 139,972.
deck <- c(rep(1:9, 4), rep(10, 16))
acceptable <- function(x) {sum(x) <= 21}
sum(combn(deck, 5, acceptable))
| {
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"url": "https://math.stackexchange.com/questions/2183749",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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k-connectedness of simplicial complexes Let $\mathcal{C}$ be a finite simplicial complex. Is there an algorithm, that can tell wether $\mathcal{C}$ is simply connected? If not, are there restrictions to $\mathcal{C}$ under which such an algorithm exists?
| No, there is no such algorithm. Here's why.
Given a finite simplicial complex $\mathcal{C}$ and a choice of vertex $v$, there is an algorithm to write down a finite presentation of $\pi_1(\mathcal{C},v)$. Conversely, given a finite group presentation $\langle g_i \,|\, r_j\rangle$ there is an algorithm to construct a f... | {
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evans book, laplace operator, estimate $\newcommand{\dd}{\text{d}}$
In the proof of symmetry of Green's function, Evans uses this estimate.
Green's function is defined as $G(x,y) := \Phi(y-x) + \phi^x$, where $\Phi$ is the fundamental solution for Laplace's equation, and $\phi^x$ is the corrector function, depending on... | I believe the source of your confusion is that you have mixed up the $x$ and $y$ in Evans' proof. The estimate he claims is
$$
\left\vert \int_{\partial B(x,\epsilon)} \frac{\partial w}{\partial \nu} v dS\right\vert \le C \epsilon^{n-1} \sup_{\partial B(x,\epsilon)} |v| = o(1)
$$
where (and here is the key part)
$$
w(... | {
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"timestamp": "2023-03-29T00:00:00",
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Is there a differentiable function on $[0,\infty)$ that satisfies $y'=y^n$ and $y(0)>0$? Let $n$ be an integer greater than $1$. Is there a differentiable function on $[0,\infty)$ that satisfies $y'=y^n$ and $y(0)>0$?
My attempt: We solve the differential equation noting that
$\frac{dy}{dt}=y^n$
$\int\frac{1}{y^n}=\int... | This is the main point: whatever positive number is picked for $y(0),$ the solution of the ODE $y' = y^n$ blows up in finite time.
If $n=2$ and $y(0) = \frac{1}{W} $ with constant $W > 0,$ then
$$ y(x) = \frac{1}{W - x} $$
for $x < W.$ The solution does not extend to $+\infty$ as requested.
If $n=3$ and $y(0) = \frac... | {
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How to Re-write Function as Unit Step Function Write the function given by: $cos(t)$, $t ∈ [0, 2π)$ and $0$ otherwise, in terms of unit step functions.
This step is at the beginning of solving a Laplace Transform, which I can do, I just don't understand this initial step.
| Note that
$$f(t)=\cos(t)(u(t)-u(t-2\pi))=\begin{cases}\cos(t)&,t\in [0,2\pi)\\\\0&,\text{elsewhere}\end{cases}$$
where $u$ is defined by
$$u(t)=\begin{cases}1&,t\ge 0\\\\0&,\text{elsewhere}\end{cases}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2184166",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is Wikipedia page on Kalman Filter's wrong? I was reading the wikipedia page on Kalman filter
Snippet from wikipedia
There, when estimating the co-variance matrix, the Q matrix is used. When calculating Kalman gain, R matrix is used.
In most literature I found on Kalman filters, it is the other way around.
Here is a sn... | Thrune is using $Q$ to denote the sensor noise covariance and $R$ to denote the process noise covariance. This is stated on page 35. Wikipedia uses the reverse of those definitions, defined in this section. Neither is wrong. I would argue that Wikipedia's notation is actually more common; I have read quite a bit of lit... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2184281",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Percentage based on Profit and Loss
What was the Percentage of Discount given?
*
*23.5 % profit was earned by selling an almirah for rs 12,350.
*If there were no Discount,the earned profit would have been 30%
*The cost price of the almirah was rs 10,000
for these Question option are
*
*only I ... | In question first statement you found cost price. But you are not able to find marked price. As for offered discount you need to know marked price.
Now placing things in an order -
From statement I -
After discount original SP 12350.
We can also calculate CP from it. CP = 10000.
From statement II -
MP = 13000 (Also SP ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2184399",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Can you calculate $\frac{dx}{dy}$ by finding $\frac{dy}{dx}$ and flipping it over? This may be a silly question but for example if you had the gradient at $x=4$ of $y=x^2+1$, then can you just calculate $\frac{dx}{dy}$ by finding $\frac{dy}{dx}$ and flipping it over? Or must you make $x$ the subject and differentiate?
| implicit differentiation operator d/dy
$y = x^2 + 1$
$d/dy(y) = d/dy(x^2 + 1)$
$1 = 2x dx/dy$ (by the chain rule on the RHS)
$dy / dx = 2x$
as expected
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2184518",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$4n$ is a square modulo $d$ implies $n$ is a square modulo $d$ I was wondering if someone could help me on a small detail that I need to clarify. Let $d$ be a squarefree integer and $n$ be any integer. Then I want to show the following:
$4n$ is a square modulo $d$ $\Rightarrow$ $n$ is a square modulo $d$. Namely if $\... | Proof for odd $d$ :
Suppose, $4n$ is a square modulo $d$, in other words $$x^2\equiv 4n\mod d$$ for some $x\in \mathbb Z_d$.
Since $d$ is odd, there exists an $y\in\mathbb Z_d$ with $2y\equiv 1\mod d$, and we have $$(xy)^2=x^2y^2\equiv 4y^2n\equiv n\mod d$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2184615",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Prove that $a * b = a + b - ab$ defines a group operation on $\Bbb R \setminus \{1\}$ So, basically I'm taking an intro into proofs class, and we're given homework to prove something in abstract algebra. Being one that hasn't yet taken an abstract algebra course I really don't know if what I'm doing is correct here.
P... | The statement of the associativity condition is wrong; it should be
$$(a \ast b) \ast c = a \ast (b \ast c) .$$
Expanding the l.h.s. gives
\begin{align}(a \ast b) \ast c &= (a + b - ab) \ast c \\ &= (a + b - ab) + c - (a + b - ab) c \\ &= a + b + c - bc - ca - ab + abc .\end{align}
Now, do the same thing for the r.h.s.... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Bounded functions and Montel's Theorem Let $\Omega\subset\mathbb{C}$ be a region and $(z_n)_n\subset\Omega$ such that $z_n\rightarrow z\in\Omega$. Also, let $f_n:\Omega\rightarrow\mathbb{C}$ be a sequence of $f_n\in\mathcal{H}(\Omega)$ such that there exists $M$ satisfying that for each $n\in\mathbb{N}$, $|f_n|<M$ and ... | Let $K \subset \Omega$ be a compact. Without LOG, we can suppose that $K$ contains an open disk centered on $x$.
Now let's proceed by contradiction, i.e. that $\vert f_n\vert $ doesn't converge uniformly to $M$. This means that it exists a real $0 < M^\prime < M$ and a strictly increasing sequence of integers $(n_j)_j$... | {
"language": "en",
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Containing an open set = being an open set? In stats courses - particularly the Casella and Berger text - reference is made to theorems being satisfied if and only if some $\Theta \subset \mathbb{R}^p$ "contains an open set," $p \geq 1$.
Isn't this just the same as $\Theta$ being an open set? Suppose there were some $\... | No it is not means the same, every subset contains an open subset that this the empty set, but is not always open like the closed ball. For the classical topology on $R$, you can take the example of the closed interval $[a,b]$ which contains the open interval $(a,b)$ but the closed interval is not open.
| {
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find a conformal map which maps a strip with a slit to the same strip without slit
find a conformal map which maps $\left\{ z: 0<\operatorname{Im z}<1\right\}$ minus $[a,a+hi]$ into the same strip without slit, where $a\in \mathbb{R}$ and $0<h<1$.
since the problem asks to eliminate slit I want to do $z^2, \sqrt{z}$ ... | If you could find a map which maps $\{z:0<\operatorname{Im} z<1\}\text{ minus }[a,a+hi]$ onto the upper half plane, then consider $$
z\mapsto \frac{1}{\pi}\log z,$$
which maps the upper half plane onto $\{z:0<\operatorname{Im} z<1\}$.
| {
"language": "en",
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Let $g(x)=\sqrt[3]{x}$ prove that g is continuous at c= 0 Please check my proof
Let $\epsilon >0$ and $\delta >0$
$$|x-0|<\delta \leftrightarrow |\sqrt[3]{x}-\sqrt[3]{0}|<\epsilon $$
$$ \leftrightarrow \sqrt[3]{x}<\epsilon $$
$$ \leftrightarrow x<\epsilon ^{3}$$
choose $\delta =\epsilon ^{3}$... | You've done most the work correctly. But some comments on your writeup:
*
*The first block of work between “Let $\epsilon > 0$” and “Choose $\delta = \epsilon^3$” is scratch work. It shouldn't be included in your final product.
*You write “Let $\epsilon > 0$ and $\delta > 0$” at the start; this is not idiomatic. ... | {
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Show that $\lim_{n\to \infty}\sin{n\pi x} =0$ if $x\in \mathbb{Z},$ but the limit fails to exist if $x\notin \mathbb{Z}.$ Show that $\lim_{n\to \infty}\sin{n\pi x} =0$ if $x\in \mathbb{Z},$ but the limit fails to exist if $x\notin \mathbb{Z}.$
1st part
If $x\in \mathbb{Z}$ then $\sin{n\pi x}=0$ for all $n,$ giving the... | For $x \in \mathbb{R}$, for $n \in \mathbb{N}$, we have $$\sin ((n+1) \pi x) - \sin (n \pi x) = \sin (n \pi x) \big( \cos(\pi x) - 1) + \cos(n \pi x)\sin(\pi x).$$
Denote $A = \cos(\pi x)-1$ and $B = \sin(\pi x)$. We have $A \neq 0$ and $B \neq 0$, and $$\sin \big( (n+1) \pi x \big) - \sin (n \pi x) = A \sin(n \pi x) +... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2185104",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Understanding smooth manifolds I'm having a little trouble understanding smooth manifolds.
I have been told the ellipsode $E = x^2+2y^{2}+3z^{2}=6$ Where $(x,y,z) \in\mathbb{R}^3$ is a 2-dimensional manifold of smoothness but I didn't really understand why. Can somebody help explain why?
Thanks
| You don't have to know what a manifold is in order to check that a subset of $\mathbb R^n$ is a submanifold!
To see that $E$ is a submanifold it suffices to check that the smooth function $f(P)=f(x,y,z)= x^2+2y^{2}+3z^{2}-6$ has a non-zero gradient at all $P\in E$, which is clear since $\operatorname {grad} f(x,y,z)=(... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Solve $f^2(x)=\frac{k}{f''(x)}$ While studying physics I have many time stumbled upon forces that are directly affected by the position of a particle. Moreover given $f(x)$ how could we approach solving the equation:
$$f^2(x)=\frac{k}{f''(x)}$$
I am new to differential equations and I would really appreciate if someone... | Putting $y = f(x)$, we have
$$y^{\prime\prime} = \frac{k}{y^2}.$$
Now multiplying both sides by $2y^\prime$, we get
$$2y^\prime y^{\prime\prime} = \frac{2ky^\prime}{y^2},$$
which can be rewritten as
$$\left( \left( y^\prime \right)^2 \right)^\prime = \frac{2ky^\prime}{y^2} = \frac{\mathrm{d}}{\mathrm{d} x} \left( -... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2185284",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Can someone help me solve $z^{3}=-i$ $z$ is a complex number. I tried to solve this question by setting $z=a+bi$, but when I calculated $(a+bi)^{3}$, I found that's a little bit complicated to compute, Can someone help me teach me some easier way to solve the problem? Thanks a lot.
| $z^3=-i \leftrightarrow z^3+i=0 \leftrightarrow (z)^3+(-i)^3=0$.
The LHS factors into $(z-i)(z^2+iz-1)=0$.
Now you have $(z-i)=0 \rightarrow z=i$ and $(z^2+iz-1)=0 \rightarrow z=\frac{-i \pm \sqrt{3}}{2}$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Why isn't the unit interval $I=[0,1]$ the universal cover of $S^1$? The question is very simple, i think. But i cannot see the answer.
The fact is that a universal cover has a fiber which is isomorphic (under a choice of a point) to the fundamental group of the covered space. In this case $\mathbb{R}$ whit the exponent... | Another way to tell that the map is not a covering map is recalling that if the base is connected, then the covering has fibers of the same cardinality, which is clearly not the case (take the fiber at $0$ and at $1/2$).
Yet another way is to use the fact that a compact base space with infinite fundamental group must h... | {
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Who can prove that a triangular number cannot be a cube, fourth power or fifth power? Triangular numbers (See https://en.wikipedia.org/wiki/Triangular_number )
are numbers of the form $$\frac{n(n+1)}{2}$$
In ProofWiki I found three claims about triangular numbers. The three claims are that a triangular number cannot b... | First, notice $n$ and $n+1$ are coprime. And if the product of coprime numbers is a n-th power then both are also n-th powers. Now divide the problem into the cases where $n$ is odd and even.
$$n=2t$$
$$t(2t+1)=a^b$$
Then $t$ and $2t+1$ are b-th powers. Let $t=y^b$, $2t+1=x^b$. Then
$$x^b-2y^b=1$$
Applying the same su... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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If A and B are two mutually exclusive events with P(A) = 0.2 and P(B) = 0.3, then what is P (A and B(complement)) If A and B are two mutually exclusive events with P(A) = 0.2 and P(B) = 0.3, then what is P (A and B(complement))
I thought it would be P(B complement) = 0.7, because it is 1-0.3, and then P (A) = 0.2, so 0... | If $A$ and $B$ are mutually exclusive, then $A \subset B^c$,
Hence $P(A \cap B^c) = P(A)$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Does, $\lim\limits_{x \to +\infty } f'(x) = + \infty \Leftrightarrow \lim\limits_{x \to +\infty } \frac{{f(x)}}{x} = + \infty $? Let $f:\Bbb R \to \Bbb R$ be a differentiable function. If $\mathop {\lim }\limits_{x \to + \infty } \frac{{f(x)}}{x} = + \infty $, it is always true that $\mathop {\lim }\limits_{x \to + ... | If $\lim_{x\to \infty}f'(x)$ exists, then from L'Hospital's Rule we have
$$\lim_{x\to \infty}\frac{f(x)}{x}=\lim_{x\to \infty}f'(x)$$
regardless of whether $\lim_{x\to \infty}f(x)$ exists or not (See the note that follows Case 2 of THIS ARTICLE).
Hence, if $\lim_{x\to \infty}f'(x)=\infty$, then $\lim_{x\to \infty}\frac... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is Digit-wise calculation possible? Suppose we have to do an intense calculation, like calculating $a^b$ for large $a$ and $b$. Then, instead of multiplying $a$ by itself $b$ times, could we just do some shortcut method with $a$ and $b$ which gives us the units digit of $a^{b}$, then another algorithm which gives the t... | If $a$ is coprime to $10$, then $a^n \mod 10^k$ is periodic in $n$ with period dividing $\varphi(10^k) = 4 \times 10^{k-1}$. Thus the lowest $k$ decimal digits of
$a^n$ are the same as those of $a^m$ where $n \equiv m \mod 4 \times 10^{k-1}$.
For example, since $21 \equiv 1 \mod 4$, the lowest digit of $7^{21}$ is th... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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What is the sigma algebra of cylindrical sets? This is a basic question but still, let $C$ be the space of real-valued continuous functions $f$ on $[0,t]$. Then a cylindrical subset of $C$ is defined as a set of the form
$$
S = \{\, f\in C; \,(f(t_1),\dots\,f(t_n))\in B\}
$$
where $B\in \mathcal{B}^n$ and $0<t_1<\dots<... | Be careful: the (or maybe "a") family of cylinder sets need not to be a priori closed under unions! You gave the right definition of cylinder set, but note that the family:
$$ \{ C_{t_1 \dots t_n} (B) : B \in (\mathcal{B})^n\, \ n \in \mathbb{N}, \ t_1 \dots t_n \in [0,T] \} $$
is just a $\pi$-system (i.e, it is close... | {
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"timestamp": "2023-03-29T00:00:00",
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Looping through zeroes in polynomials Question.
If $p$ is a polynomial of degree $n$ with $p(\alpha)=0$, what do we know of the polynomial $q$ (with degree $n-1$) such that the numbers $(q^k(\alpha))_{k=1}^n$ contain all of the zeroes of $p$?
Here I denote $q(q(\cdots q(\alpha)))=q^k(\alpha)$.
Notes.
We know for a f... | Let $\alpha=\alpha_1, \alpha_2, \ldots, \alpha_m$ be the distinct roots of $p$.
Choose a permutation $\sigma$ of $1,2,\dots,m$ without fixed points. For instance, an $m$-cycle such as $(12\cdots m)$.
Let $q$ be the unique polynomial such that $q(\alpha_i)=\alpha_{\sigma(i)}$. That will work, but won't have degree $n-1$... | {
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"url": "https://math.stackexchange.com/questions/2186076",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Prove that if $r(t)\times \frac{dr(t)}{dt}=0$, then $r(t)$ has a fixed direction. Let $r(t)$, where $t$ is a parameter $(t ∈R)$, be a position vector such that $r(t)\times \frac{dr(t)}{dt}=0$
I am asked to show that r(t) has a fixed direction.
A hint says: Let $r(t) = f(t)\hat e(t)$ where $\hat e(t)$ is a unit vector). ... | If $r(t)=f(t)ê (t)$ then
$$ \frac{d \vec r}{dt} = f'(t)ê (t) + f(t)\frac{dê (t)}{dt} $$
$$ \vec r \times \frac{d \vec r}{dt} = \vec r \times f'(t)ê (t) + \vec r \times f(t)\frac{dê (t)}{dt} = \vec r \times f(t)\frac{dê (t)}{dt} = 0$$
Thus $\frac{dê (t)}{dt} = 0$ or $\vec r$ is parallel to $\frac{dê (t)}{dt}$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2186197",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Injectivity of the factorial-related map $f:\mathscr P(\Bbb N) \to \Bbb R$, $f(I) = \sum_{n \in I} \frac 1 {n!}$ Note: For the purpose of this question, $\Bbb N$ does not include $0$.
I have a function $f:\mathscr P(\Bbb N) \to \Bbb R$ defined by:
$$f(I) = \sum_{n \in I} \frac 1 {n!}$$
This is essentially a transformat... | Hint First prove that $f(\{k\}) > f(\{k + 1, k + 2, \ldots\})$ for all $k \in \Bbb N$.
Then, consider distinct elements $I, J \in \mathscr{P}(\Bbb N)$ and the smallest $k \in \Bbb N$ which is in one and not the other. (You'll probably also want to use the apparent fact that $f$ is monotonic under inclusion, that is, th... | {
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Can we define a 'rectangular coordinate' on a curved surface? We use rectangular coordinate on a flat plane, so can we use it in a curved surface, like the axis is somehow bent?
If yes, is there any application? Also, can we generalize this to higher dimension?
| Assume we have a parametrized surface (patch) given by $S:[0,1]^2\rightarrow \mathbb R^n$ and we are looking for a reparametrization function $\varphi:[0,1]^2\rightarrow [0,1]^2$ so that
$$\langle \partial_x (S\circ \varphi),\partial_y (S\circ \varphi)\rangle=0,$$
which expresses this perpendicularity claim you are loo... | {
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"timestamp": "2023-03-29T00:00:00",
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In mean value theorem over $[x_0,x]$, will $x^* \to+\infty$ always as $x \to +\infty$? The mean value theorem states that if $f:\Bbb R \to \Bbb R$ is continuous over $[x_0,x]$, and differentiable over $(x_0,x)$, then there exists $x^*\in(x_0,x)$ s.t. $f'({x^*}) = \frac{{f(x) - f({x_0})}}{{x - {x_0}}}$.
Now suppose $f$ ... | This is false. Consider $f\left( x \right) = \left\{ {\begin{array}{*{20}{c}}
0&{x \geqslant 0} \\
{{e^{\frac{1}{x}}}}&{x < 0}
\end{array}} \right.$
It is easy to verify $f(x)$ is differentiable over $\Bbb R$ and ${f'}\left( x \right) = \left\{ {\begin{array}{*{20}{c}}
0&{x \ge 0}\\
{ - {x^{ - 2}}{e^{{x^{ - 1}}}}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2186573",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How does a function satisfy the Lipschitz condition? Currently taking a higher level differential equations class and we're studying existence and uniqueness of solutions. Multiple times during proofs, my professor uses Lipschitz to say $$f(t,x)-f(t,y)\le L(x-y)$$ This concept makes sense to me as it only works if a fu... | There is no general criterion besides the definition, but there are some known results. The easiest one is to check whether $f$ is differentiable with bounded derivative, in which case it is Lipschitz with constant $||df||$, which is an immediate consequence of the mean value theorem (at least in convex regions). Often... | {
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Does $\int _0^{\infty }\:\frac{1}{1+x^2\left(\sin x\right)^2}\ \operatorname dx$ converge? I have been trying to prove the following integral:
$$\int _0^{\infty }\:\frac{1}{1+x^2\left(\sin x\right)^2}\ dx$$
diverges (please correct me if I am mistaken).
I have tried to use different comparison tests (as this is an ... | The idea is to bound the integral below on intervals where $\displaystyle \frac{1}{1+x^2\left(\sin x\right)^2}$ has spikes, that is to say, it suffices to find some $\varepsilon_k$ such that $$\sum_{k\geq1}\int_{k\pi -\varepsilon_k}^{k\pi +\varepsilon_k}\frac{1}{1+x^2\left(\sin x\right)^2}dx$$ diverges.
On each of thes... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Curve sketching: Desmos shows an oblique, absolute value asymptote I sketched the function $f(x) = x^{6/7}-9x^{2/7}$ and got something like this.
Where POI means point of inflection.
However, when I graph it in Desmos, I get what looks like an oblique asymptote, that corresponds to an absolute value function.
The mor... | In fact, there is no contradiction concerning your POI, with abscissas at $\pm (15)^{7/4} \approx \pm 114.3$ : it is impossible to spot them even on an large curve plainly because the transition from positive to negative concavity is very faint.
See graphics below obtained with Geogebra. The first one for the variation... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2186902",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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How many four-letter words w using an n-letter alphabet satisfy... This question has two parts. The first part was easy.
a) How many four-letter words w using an n-letter alphabet satisfy $w_i \neq w_{i+1}$ for $i=1,2,3$
Simple. Choose the first letter in $n$ ways. Because no consecutive letter can have the same letter... | Your answer to the first question is correct.
You have also correctly identified the cases in the second question.
Case 1: The third letter is the same as the first letter.
We have $n$ ways to select the first letter. Since the second letter must be different from the first, we can select it in $n - 1$ ways. We have... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2187067",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find the limit of a function as $ x$ approaches $0$ How can I find the limit of $\dfrac{\cos(3x) - 1 }{x^2}$ as $x\to 0$?
If someone could please break down the steps, for clear understanding. I'm studying for the GRE. Thanks in advance !!
| Since you are studying for the GRE, which if I recall correctly is a multiple choice exam, knowing L'Hospital's rule is a good thing. Here's the simple version:
When you plug in your value that $x$ is approaching, if you end up with $0/0$ or $\infty/\infty$, then you can take the derivatives of the top and bottom, and ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2187175",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
} |
Let $f : \Bbb R \rightarrow \Bbb R$ be a func such that $p>0$, that $f(x+p) = f(x)$ for all $x \in \Bbb R$ . Show that $f$ has an absolute max and min Problem: Let $f : \Bbb R \rightarrow \Bbb R$ be a contiunous function such that for some real number $p>0$, $f(x+p) = f(x)$ for all $x \in \Bbb R$. Show that $f$ has an ... | Theorem: Any continuous map on a closed interval has a max and a min.
Apply this to the interval $[0,p]$. Since for any $x \in R$ we have some $x_0 \in [0,p]$ such that $f(x)=f(x_0)$, the maximum on that sub-interval is in fact a global max.
Note: No differentiability assumption was made.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2187278",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Decidability of regularity of context-free grammar I've searched for a long time, but cannot find this. Maybe it's an open problem, but it seems not that hard.
Let's say I have a context-free grammar, say in Chomsky normal form for definiteness. Is there an algorithm to check whether it generates a regular language?
Of... | The following theorems are proved in Jeffrey Shallit's A Second Course in Formal Languages and Automata Theory.
Theorem 6.6.6. It is undecidable whether, given a CFG $G$, $L(G)$ is regular.
Theorem 6.6.7. There exists no algorithm that, given a CFG $G$ such that $L(G)$ is regular, outputs a DFA that accepts $L(G)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2187431",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
solve this diophantine equation: ${\rm lcm}[x,y]+{\rm lcm}[y,z]+{\rm lcm}[z,x]=3(x+y+z)$ I am almost certain it is a duplicate question but I am looking for a reference regarding how solve the diophantine equation，Find the postive integer $x,y,z$ such
$${\rm lcm}[x,y]+{\rm lcm}[y,z]+{\rm lcm}[z,x]=3(x+y+z)$$
| Now that's a nice little problem. I didn't expect it to be solvable completely, but... wait a minute.
Obviously, a multiple of any solution is also a solution, so we may just as well divide it by $\gcd(x,y,z)$, if any, and look for the primitive triples. Now, being coprime as a triple does not mean being pairwise copri... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Discrete math(Divisors and primes) Is the following statement true or false?Explain.
There are integers $x,y,$ and $z$ such that $14$ divides $2^x × 3^y × 5^z$.
My guess is false but I don't know how to explain it?Does it have anything to do with the fundamental theorem of arithmetic?
| Here's an intuitive explanation
Yes $14$ does not divide $2^x × 3^y × 5^z$ for any integers $x,y,z$ because
$14=2 × 7$
Since there are no seven's in the expression..It will never fully divide it
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2187676",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Minimize $A=\frac{1+2^{x+y}}{1+4^x}+\frac{1+2^{x+y}}{1+4^y}$ For $a,b>0$. Minimize $$A=\frac{1+2^{x+y}}{1+4^x}+\frac{1+2^{x+y}}{1+4^y}$$
i think we let $2^x=a;2^y=b$
Hence $A=\frac{1+ab}{1+a^2}+\frac{1+ab}{1+b^2}$
We need pro $A\geq 2$(Wolfram Alpha) but $x,y$ is a very odd number and i can't find how to prove it $\g... | we have to prove that $$(1+ab)\left(\frac{1}{1+a^2}+\frac{1}{1+b^2}\right)\geq 2$$ and this is equivalent to $${\frac { \left( ab-1 \right) \left( a-b \right) ^{2}}{ \left( {a}^{2}
+1 \right) \left( {b}^{2}+1 \right) }}
\geq 0$$ this is right if $$ab\geq 1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2187751",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
maximum number of number of roots of $p(x) = 0$ is
Let $p(x)=x^6+ax^5+bx^4+x^3+bx^2+ax+1.$ Given that $x=1$ is a root of $p$ but $x=-1$ is not, find the maximum number of number of roots of $p$.
My attempt:
$x=0$ in not a root of $p(x)=0.$ So
$$\left(x^3+\frac{1}{x^3}\right)+a\left(x^2+\frac{1}{x^2}\right)+b\left(x+\... | This answer assumes that you want to find the maximum number of the real roots of $p(x)$.
You already have
$$(t-2)\left(t^2+(a+2)t+a-\frac 12\right)=0$$
where $t=x+\frac 1x$ (which is correct though you have a typo in the part $a(x^2+\frac{1}{x^2})=a(x+\frac 1x)^2-2\color{red}{a}$).
Let $t_{\pm}$ where $t_-\lt t_+$ be ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2187861",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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Are these funcitons linearly independent? Let $a,b,c,\ldots$ be a finite set of distinct positive real numbers.
Are the functions
$(a+x)^{-r}$, where $r$ is a positive real number, linearly independent function on $[0,\infty)$? Are there any references for this? Would the answer depend on r?
| Let $0 < a_1 < \dots < a_n$ and assume that $\left( (a_i + x)^{-r} \right)_{i=1}^n$ are linearly dependent over $[0,\infty)$ and so we can find $b_1,\dots,b_n \in \mathbb{R}$ with
$$ \sum_{i=1}^n \frac{b_i}{(a_i + x)^r} = \sum_{i=1}^n \frac{b_i}{e^{r \ln(a_i + x)}}= 0 $$
for all $x \in [0,\infty)$. Note that the funct... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2187994",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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} |
Prove that trigonometric equation isn't changed by value of x $\sqrt{\sin^4x+\cos2x} + \sqrt{\cos^4x-\cos2x}$ I have to prove that $x$ doesn't matter. However I can't get things to simplify.
| Hint. One has
$$
\sin^4x+\cos2x=\sin^4x+2 \cos^2x-1=\sin^4x-2 \sin^2x+1
$$ and
$$
\cos^4x-\cos2x=\cos^4x-2 \cos^2x+1
$$ Can you finish it?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2188149",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Integration by Substitution I have the following problem: $\int (x+2)(2x-3)^6dx$
What I did was allow $u = 2x-3$
$dx= \frac{1}{2}du$
Since I am integrating w.r.t u I decided to let $x=\frac{u+3}{2}$
my new equation is $\int ( \frac{u+3}{2}+2)(u^6)\frac{1}{2}du$
which I then simplify to $\int ( \frac{u+7}{4})(u^6)du$
I ... | simplifying your Integrand we get $$\frac{1}{4}\int (u+7)u^6du$$ further we get the integral $$\frac{1}{4}\int\left( u^7+7u^6\right)du$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2188269",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Proving $\int_0^\infty\frac{\sin(x)}x\ dx=\frac{\pi}2$. Why is this step correct? I came across a different approach on the proof: $$\int_0^\infty \frac{\sin(x)}x\ dx=\frac{\pi}2$$
First, recall the identity: $$\sin(A)-\sin(B)=2\sin\left(\frac{A}2-\frac{B}2\right)\cos\left(\frac{A}2+\frac{B}2\right)$$
Applying the ide... | Let $l_n=n/\pi-1/2$ for $n/\pi>1/2$. Then $$\int_0^n t^{-1}\sin t\;dt =\int_0^{\pi} t^{-1}\sin (l_nt+t/2)\;dt.$$ Now let $n\to \infty.$
It would have been clearer if this had been said .
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2188410",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 1
} |
isosceles triangle height and base given only angles and area If we know only that:
$A=22$
$v_1=30^\circ$ and $v_2=75^\circ$ and $v_3=75^\circ$
How do I get height and base without knowing at least one of the sides?
Edit: v are angles and A is the area.
|
We have that $A$ = $22$
Let $M$ be the midpoint of $AC$. Then $A = 2\cdot \frac{1}{2}\cdot AM \cdot MB = AM\cdot MB$
You also have the relationship $AM = AB\cos(75^{\circ})$ and $AC = 2AM$.
Let $AB = a \Rightarrow BC = a$
Let $AM = b$
Then we have the area of the triangle = $22 = \frac{1}{2}$ X base X height= $b\sqrt{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2188556",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Is the set of diagonal matrices with positive entries open in the set of positive definite symmetric matrices? I suspect that it's not, but would like to know a proof for why the set of diagonal matrices with positive entries is or isn't open in the set of positive definite symmetric matrices.
I am familiar with what ... | Let $A$ be diagonal with positive diagonal entries, choose as $B$ any positive definite matrix that is not diagonal and consider $A_n := A + \frac B n$. Then each of the $A_n$ is positive definite and none of them is diagonal. So, your set is definitely not open (except for the case where the space is one-dimensional).... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2188634",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Irreducibility of polynomials in ring In text book "A course in abstract algebra" by author Khanna & bhambri" it is given that,
f(x) = 2( x^2) + 2 is irreducible polynomial over Z.
Because they used the definition "let R be an Integral domain with unity then a polynomial f(x) in R[x] of positive degree (i.e. deg ≥ 1... | To dramatize the flaw in the definition given in the text by "Khanna & Bhambri" (K &B), consider the polynomials
$$x,\;2x,\;3x,\;6x$$
By K&B's definition, the above polynomials are all irreducible in $\mathbb{Z}[x]$. Moreover, since none of them is a unit factor times one of the others, they would be regarded as distin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2188830",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Does $M^{*}$ contain an injective function?
Let $R$ be a PID and $M$ a finitely generated $R$-module. Let $F$ be the field of fractions of $R$ and $P = F/R.$ Let $M^{*}$ = $\mathrm{Hom}_R(M, P).$ Suppose $M$ is torsion. Is there an injective function in $M^*?$
What I've done so far was let $M =\left <e_1, ..., e_n\r... | The answer is:
$M^*$ contains an injective morphism if and only if $M$ is a cyclic
torsion $R$-module, i.e. $M \cong R/(a)$ for some $0 \neq a \in R$.
The backward direction is trivial, because $R/(a) \cong \langle \frac{1}{a} \rangle \subset F/R$.
The forward direction goes as follows: It is well known that any fi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2188927",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Graph Theory proving planarity I have these set's of graphs:
I used Euler's inequality, and the 4 color theorem which resulted in a inconclusive result. Using Kuratowski's theorem I was unable to create a K3 3 or K5 graph. So does this prove that the graphs are planar? How do we really know that there does not exist ... | Graph $G$ is not planar. The subgraph you get by deleting the edges $ab,$ $ad,$ $bd,$ $cg,$ $eg$ is homeomorphic to $K_{3,3};$ the vertices $a,b,d$ are connected to the vertices $e,f,g$ by the internally disjoint paths $ace,$ $af,$ $ag,$ $be,$ $bf,$ $bg,$ $de,$ $df,$ $dg.$
Graph $H$ is planar. Plot $a$ at $(0,0),$ $b$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2189034",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 0
} |
What is a Universal Construction in Category Theory? From pg. 59 of Categories for the Working Mathematician:
Show that the construction of the polynomial ring $K[x]$ in an indeterminate $x$ over a commutative ring $K$ is a universal construction.
Question: What does the author mean by this bolded term?
For context: ... | A universal construction is simply a definition of an object as "the unique-up-to-isomorphism object satisfying a certain universal property". The name "construction" is a little misleading, because it's a definition: one still needs to show that the object actually exists, and that usually has to be done non-category-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2189124",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 2,
"answer_id": 1
} |
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