Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
If there is a monomorphism between free $R$-modules $F \hookrightarrow G$, is the rank of $F$ necessarily less than or equal to the rank of $G$?
Let $R$ be a commutative, unital ring. Let $F = R^{(X)}$ and $G = R^{(Y)}$ be the free $R$-modules generated two arbitrary sets $X$ and $Y$ respectively. Assume there is a mo... | The infinite case follows from the finite case. Specifically, supposing $Y$ is infinite, for each finite subset $S\subseteq Y$, let $A_S\subseteq X$ be the set of basis elements of $F$ which map to elements of $G$ whose support is contained in $S$. Note that our monomorphism $F\to G$ then restricts to a monomorphism ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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A Quick, Calculus II-Level Proof Regarding the Divergence of Certain Series? I came across the following problem in the Calculus text I'm using to teach my class:
Suppose $\sum a_n$ is a positive series such that $\{a_n\}$ is a decreasing sequence and the sequence $\{na_n\}$ converges, but not to zero. Show that $\sum ... | If $na_n\to c\ne 0$ use limit comparison with $b_n = c/a_n$. (By the way, the decreasing hypothesis is not needed.)
For the second one, direct limit comparison is obvious. I’m not sure what they had in mind.
| {
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"timestamp": "2023-03-29T00:00:00",
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Fuglede's theorem in finite-dimensional vector space Let $V$ be a finite dimensional vector space and $A$ be normal operator on $V$ and $B$ is an operator such that $AB=BA$. Show that $BA^*=A^*B$.
I guess that this problem should not be so difficult. I have tried different approaches and I got some identities which do ... | The way to think about this problem is when $B$ is diagonalizable, and $A$ being normal is diagonalizable (over $\mathbb C$) so we can call on simultaneous diagonalizability, recognize that being normal $A^*$ may also be simultaneously diagonalized with $B$ (via the same similarity transform that we'd use on $AB$) whi... | {
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"timestamp": "2023-03-29T00:00:00",
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Functional equation $f(x+1)=af(x)+b$
Functional equation $f(x+1)=af(x)+b$
There was a question I solved a few days back that asked for a closed form of an equation for a given system. The function came down to this equation which I solved by noting the pattern. By the way $f(0)=10$
So here's how I saw it:
$f(1)=10a+b... | $$f(x+1)-af(x)=b~~~~(1)$$
Jet $f(x)=g(x)+c$, then
$$g(x+1)+c-ag(x)-ac=b$$
$$g(x+1)-ag(x)=0, c=b/(1-a)$$
Let $$g(x)=d t^x \implies t=a$$
So the solution of (1) is
$$f(x)=da^x+\frac{b}{1-a}$$
| {
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Different solutions with different results for an inequality Find m such that the following inequality:
$$\left|4x-2m-\frac{1}{2}\right| > -x^2 +2x + \frac{1}{2} - m$$
is always true for $\forall x \in R$.
1st solution:
1st case
$$4x-2m-\frac{1}{2} > -x^2 + 2x +\frac{1}{2} -m$$
$$<=>x^2+2x-m-1>0$$
$$\Leftrightarrow 1^2... | Your mistake in the first solution: at some stage you drop the variable $x$ by using $\forall x\in\mathbb R$. But as you are doing this case analysis, the $\forall$ no longer holds.
Your mistake in the second solution: from $a>0, a>b$, you conclude $0>b$, which is not tight. (Take $a=2,b=1$.)
| {
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Finding radius of curvature of a catenary with length of string given I had the following question, asking about the radius of curvature (in$\text{ cm}$) of the string, at the bottom most point, fixed at two ends, with the angle with the vertical being $\pi/4$ at the fixed points, and length of string $40 \text{ cm}$.
... | Using your equations
Hoping that I properly understood, we have
$$y(x)=\int_{0}^{x}\sqrt{1+a^2\sinh^2(t)}\,dt=20$$
Making $t=i u$
$$\int\sqrt{1+a^2\sinh^2(t)}\,dt=i \int\sqrt{1-a^2\sin^2(u)}\,du=i E\left(u\left|a^2\right.\right)=-i E\left(i t\left|a^2\right.\right)$$
$$y(x)=-i E\left(i x\left|a^2\right.\right)=20$$
$$\... | {
"language": "en",
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"source": "stackexchange",
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Prob. 4, Sec. 29, in Munkres' TOPOLOGY, 2nd ed: $[0, 1]^\omega$ with uniform topology is not locally compact Here is Prob. 4, Sec. 29, in the book Topology by James R. Munkres, 2nd edition:
Show that $[0, 1]^\omega$ is not locally compact in the uniform topology.
Here is a Math Stack Exchange (MSE) post that is of co... | I think the proof is fine, and easy enough. It's a generalisation of the idea to show the unit ball in $\ell^\infty$ not being compact.
Such infinite-dimensional linear-like spaces will almost never be locally compact.
| {
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Solution of a first order non linear differential equation I am trying to find the extrema of the integral below
$$
I= \int_0^1 y^2 \mathrm dx
$$
under the conditions
$$
\int_0^1 \left(\frac{dy}{dx}\right)^2 \mathrm dx =1
$$
and y(0)=y(1)=0
Using Lagrange multiplier λ, equivalently, I can find the extrema of the below... | The Beltrami equation can not contain a second derivative, where should it come from? It should evaluate to
$$
y^2-λy'^2=C
$$
which easily leads to solutions in terms of trigonometric or hyperpolic functions.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3612565",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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The value of $\lim_{n\rightarrow \infty}(-0.5)^n$ is
The value of $\lim_{n\rightarrow \infty}(-0.5)^n$ is
What i have tried
As i know $\lim_{n\rightarrow \infty}(a)^n=0$ when $0<a<1$
If it is $\lim_{n\rightarrow \infty}(0.5)^n$ . Then it is $0$
But How do i solve for negative exponent. Help me plese
| $-((0.5)^n) \le (-0.5)^n \le (0.5)^n.$
Squeeze.
| {
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$f: X \to Y$ order preserving implies $Ord(X) \leq Ord(Y)$ Let $X,Y$ be well-ordered sets and let $f:X \to Y$ be a map that preserves the strict order. I would like to prove that $Ord(X) \leq Ord(Y)$.
You can assume I know the 'basic' results about maps on well-ordered sets and segments etc.
I feel like there should be... | If $\beta<\alpha$, you can view $f$ as a map from $\alpha$ into $\alpha$. $\{\xi\in\alpha:f(\xi)<\xi\}\ne\varnothing$, since clearly $f(\beta)<\beta$. Let $\eta=\inf\{\xi\in\alpha:f(\xi)<\xi\}$, and derive a contradiction with the assumption that $f$ is strictly order-preserving.
| {
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"timestamp": "2023-03-29T00:00:00",
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Prove $\forall xA(x) \to B \therefore \exists x(A(x) \to B)$. Working on P.D. Magnus. "forallX: an Introduction to Formal Logic" (p. 297, exercise C. 1):
$
\def\fitch#1#2{\quad\begin{array}{|l}#1\\\hline#2\end{array}}
\def\Ae#1{\qquad\mathbf{\forall E} \: #1 \\}
\def\Ai#1{\qquad\mathbf{\forall I} \: #1 \\}
\def\Ee#1{\q... | You are quite correct; that is invalid.
Try building the proof with these assumptions.
$$\def\fitch#1#2{\quad\begin{array}{|l}#1\\\hline#2\end{array}}\fitch{~~1.~\forall x~A(x)\to B}{\fitch{~~2.~\neg\exists x~(A(x)\to B)}{\fitch{~~3.~A(b)}{\fitch{~~4.~\neg A(a)}{\fitch{~~5.~A(a)}{~~\vdots}\\~~\vdots}\\~~\vdots}\\~~\vdo... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Showing that $\tau(t) = (t^2, t^3)$ is not a submanifold
Let $\tau : \mathbb{C} \to \mathbb{C}^2$ be the map $\tau(t) := (t^2, t^3)$. Show that $\tau$ defines an embedding map from $\mathbb{C}^*$ to $\mathbb{C}^2 \setminus{0}$. Is $\tau(\mathbb{C})$ a submanifold of $\mathbb{C}$?
Note that the definition of an embedd... | HINT: if it were a submanifold then around $(0,0)$ it would be the graph of a function of $x$ or a function of $y$. Check neither is true.
| {
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Is my proposition correct? I have concluded (proved) the following, but haven’t seen it stated anywhere, hence would like to get it verified by you mathematicians.
Let $(a_n)_{n=m}^\infty$ be a sequence of reals for some $m\in\mathbb{Z}$ such that $(|a_n|)_{n=m}^\infty$ converges to some $L\in\mathbb{R}$. (Hence, $L\g... | (Yes it's true) Hint : At least one of the two sets $A_+ := \{n : a_n > 0\}$ and $A_- := \{n : a_n \leq 0\}$ is infinite. W.L.O.G suppose that $A_+$ is infinite; then it immediately follows that $L$ is a limit point of $(a_n)_n$. And therefore $$\varlimsup (a_n)_n = \varlimsup_{n \in A_+} (a_n)_n = \lim_{n \in A_+}(a_n... | {
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What is the range of the function $\frac{3}{2-x^2}$ I'm so, so very confused about finding the ranges of real functions, no concept in Mathematics has yet confused me more than this, please tell me what's wrong in my solution for finding the range of the function : $\dfrac{3}{2-x^2}$
Here's how I do it and get a partia... | We have: $$y=f(x)=\frac{3}{2-x^2} ; x \ne \pm\sqrt{2}$$
Write $x$ as a function of $y$ as: $$x=\sqrt{2-\frac{3}{y}}$$
Now we have to find the set of values of $y$ for which $x$ is $\mathbf {real}$.
For this, $2-{3\over y} \ge 0$. This can be acquired in two ways:
$\mathbf {Case\ 1:}$ Substituting $x=0$ into the second... | {
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Series Trouble / Diverging Does anyone mind explaining both of these questions, I’m stuck.
*Determine whether the sequence converges or diverges. If it converges, find the limit it converges to. $$\left\lbrace\sqrt[n]{2^{1 + 2n}}\right\rbrace_{n = 1}^\infty$$
*Find the general $n^{th}$ term of the sequence. Then d... | The second one (problem 4) seems to be $u_n = \frac{n^2}{n!}$ and as $$0 \le u_n \le \frac{n}{n-1}\frac{1}{n-2} \le \frac{4}{n}$$
for $n \ge 4$, it converges to zero.
| {
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"timestamp": "2023-03-29T00:00:00",
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For $f\in L^1(\mathbb{R})$, show that $\lim_{\varepsilon \to 0}\int_{-\infty}^\infty \cos(\varepsilon x)f(x) \, dx=\int_{-\infty}^\infty f(x)\,dx$ Question: For $f\in L^1(\mathbb{R})$, show that
$$\lim_{\varepsilon \rightarrow 0}\int_{-\infty}^\infty \cos(\varepsilon x)f(x)\,dx=\int_{-\infty}^\infty f(x)\,dx $$
where t... | Crucially, for an $L^1$ function, most of the mass will be concentrated on a large interval and it is this same large interval where each integrand in your sequence will have most of their mass.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3614477",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prob. 6, Sec. 29, in Munkres' TOPOLOGY, 2nd ed: Is this map a homeomorphism? Let $n$ be any given natural number, and let
$$ S^n \colon= \left\{ \, \left( x_1, \ldots, x_{n+1} \right) \in \mathbb{R}^{n+1} \, \colon \, \sum_{i=1}^{n+1} x_i^2 = 1 \, \right\}. $$
Let point $\mathbf{p} \in \mathbb{R}^{n+1}$ be given by
$$ ... | Goal of this Answer
This isn't a complete solution, rather it serves as some notes to help you get over some of the humps in this analysis. I will cover
*
*Injection of $f$
*Surjection of $f$
*Obtaining $f^{-1}$
*Small Conclusion
hope you find this helpful.
Injection
Use the fact that
$$\sum_{i=1}^{n+1}u_{i... | {
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Trouble with $4\times4$ matrix determinant $$
\begin{vmatrix}
1 & -6 & 7 & 5 \\
0 & 0 & 3 & 0 \\
3 & -2 & -8 & 6\\
2 & 0 & 5 & 4\\
\end{vmatrix}
$$
Clearly I want to expand along the second row yielding:
$((-1)^5)3$ times the following matrix
$$
\begin{vmatrix}
1 & -6 & 5 \\
3 & -2 ... | Another way to approach this 4x4 matrix is to row reduce to a triangular matrix with zeros underneath the diagonal.
\begin{bmatrix}1&-6&7&5\\0&0&3&0\\3&-2&-8&6\\2&0&5&4\end{bmatrix}
Row reducing to the triangular matrix yields:
\begin{bmatrix}1&-6&7&5\\0&16&-29&-9\\0&0&3&0\\0&0&0&\frac{3} {4}\end{bmatrix}
From here, w... | {
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Constructing an equilateral triangle of a given side length inscribed in a given triangle
I am trying to solve the problem of constructing, with straightedge and compass, an equilateral triangle of given side length $a$ inscribed in a given triangle.
I found this post "Inscribe an equilateral triangle inside a triang... | I believe the following diagrams and incorporated explanation will suffice. Let me know if it is not clear. Click on image to get a larger and clearer view.
| {
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"timestamp": "2023-03-29T00:00:00",
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Good way of explaining how to find all solutions to $\sin\theta+a=0$ I'm not sure exactly how to properly describe this trigonometry problem, so I will just write it out. It is less of a "problem" and more of a "I can't find a really good way to explain it".
We want to find a value $\theta$ such that $$\sin\theta + a =... | One way of deriving this result is graphically: if we plot the graphs $y = \sin x$ and $y=a$, we see the following behaviour
It's easy to infer a pattern from this graph: $a$ equals $\sin x$ when $$x \in \{... -\pi - \arcsin a, 0 + \arcsin a, \pi - \arcsin a, 2\pi + \arcsin a ...\}$$
the pattern is thus
$$x = n\pi + (... | {
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How to convert the probability P/Q to P⋅Q−1 where Q is co-prime with mod I was dealing with probability in programming but i was stuck on the final answer part.
Below is statement in which format i have to give the answer
Can you find the probabilities It can be proved that for each of these values, the probability can... | p=998244353;
int power(int x, int y, int p)
{
int res = 1; // Initialize result
x = x % p; // Update x if it is more than or
// equal to p
while (y > 0)
{
// If y is odd, multiply x with result
if (y & 1)
res = (res*x) % p;
// y must be even now
y = y>>1; // y ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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$12$ men can finish a job in $16$ days. $5$ men work at the start; and after 8 days, 3 men were added. How many days needed to finish the whole job?
Twelve men can finish a job in 16 days. 5 men were working at the start and after 8 days, 3 men were added. How many days will it take to finish the whole job?
Solution:... | $12$ men can finish the job in $16$ days.
In $8$ days, $24$ men can finish the job.
So $5$ men would've finished $\frac 5{24}$ of the job within those $8$ days, leaving $\frac{19}{24}$ of the job remaining.
Add $3$ men, you get $8$ men now. $8$ men would take $24$ days to finish the job.
So the reinforced workforce wou... | {
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How to express matrix multiplication as the sum of individual elements? I want to express $$\sum_{i=1}^N \sum_{j=1}^N a_{ij} x_i x_j$$ as matrix/vector multiplication. I've managed to get that the expression above is equal to $$x'Ax$$ where x is a vector, by expanding the sum but it took a while. Is there a faster way,... | Yes, it depends upon what value of $a_{ij}$ and $x$ you are choosing. Relate your question with this example. Suppose $x_i = 1$ and $x_j = 1$ for every value of $i$ and $j$ and $$a_{ij} = \frac{a_i a_j }{i+j+1}$$ where $1\le i,j\le N$ and $a_i, a_j \in \Bbb N$ then $$\sum_{i=1}^N \sum_{j=1}^N a_{ij} x_i x_j = \sum_{i... | {
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Find the Fourier series for $f(\theta)=\theta^2$ and prove $\Sigma^\infty_{n=1} \frac{1}{n^2}=\frac{\pi^2}{6}$ Find the Fourier series for $f(\theta)=\theta^2$ and use Parseval's identity for $f$ to derive the identity:
$$\sum^\infty_{n=1} \frac{1}{n^2}=\frac{\pi^2}{6}$$
In addition, find the expansion for $f$ in terms... | You don't need Parseval (which will square your coefficients so it will give you something with $n^4$). From this question,
$$
f(x)=\frac{\pi^2}{3}+4 \ \sum_{n=1}^{+\infty} \frac{(-1)^n}{n^2} \ \cos(nx).
$$
As $f$ is differentiable everywhere, we have pointwise convergence. Then, evaluating at $\pi$,
$$
\pi^2=f(\pi)=\... | {
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"timestamp": "2023-03-29T00:00:00",
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Density of $X_1 + \cdots+X_n$ when $X_i$'s are independent $U(-1,1)$ variables How can we show for independent random variables uniformly distributed over $(1,-1)$ that $X_1 + \cdots+X_n$
has density
$$\pi^{-1}\int^\infty_0 \left(\frac{\sin t}{t}\right)^n \cos tx \; dt \textrm{ for }n \geq 2\text{?} $$
This is probl... | Take $Y_n:=\sum_{k=1}^nX_k$
Compute the characteristic function of $Y_n$ :
we know that $\varphi_{X_1}(x)=\frac{\sin(x)}{x}$ if $x \neq 0,$ and $\varphi_{X_1}(0)=1,$ using independence we obtain that $\varphi_{Y_n}(x)=\frac{\sin^n(x)}{x^n}$ if $x \neq 0$ and $\varphi_{Y_n}(0)=1.$
Notice that $\varphi_{Y_n}$ is an odd f... | {
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mapping class group of the real projective plane In most literature I've read about the mapping class group, I found that many authors have stated without any explanation that any homeomorphism of a real projective 2-space to itself is isotopic to the identity. I'm guessing it is obvious but I can't seems to come up wi... | The free mapping class group (that is, the path components of its full group of homeomorphisms) of the Möbius band M is not trivial. It has a self-homeomorphism h : M → M whose effect h* : H1(M;Z) → H1(M;Z) on the first homology group (which is infinite cyclic) is multiplication by -1.
The projective plane P2 can be m... | {
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ODE help with initial conditions Initial conditions are $f(0)=1, f'(0)=0$
Applying the Laplace Transform to both sides and simplifying I reach the form
$$ L(y) = \frac{s^3 + 5s}{s^2 + 4}, $$
which I then further simplify in the following steps
$$ L(y) = \frac{s(s^2 + 5)}{(s^2 + 4)(s^2 + 9)} $$
$$ L(y) = \frac{s(s^2 + ... | $$y'' + 9y = \cos(2x)$$
$$\implies r=\pm3i$$
$$y(x)=c_1 \cos 3x + c_2 \sin 3x$$
Then for the particular solution
$$y_p=A\cos (2x) \implies A=\frac 15$$
The solution is therefore:
$$y(x)=c_1 \cos (3x) + c_2 \sin (3x) +\dfrac 15 \cos (2x)$$
Apply initial conditions:
$$c_2=0 \text { and } c_1=\frac 45$$
So that:
$$y(x)=\f... | {
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Why is $f(x)g'(x)+f'(x)g(x)$ a linear map? Assume $f:E \rightarrow \mathbb{R}^m$ and $g:E \rightarrow \mathbb{R}$, and $E \in \mathbb{R}^n$ is open. Assume $x \in E$, and $f$ and $g$ are differentiable at $x$. To get the derivative of $f(x)g(x)$, I get a product rule version, namely $f'(x)g(x)+g'(x)f(x)$.
According to... | The derivative of $f$ at $x_0$ is a linear map $A$ such that $f(x) - f(x_0) - A(x-x_0) = o(x-x_0)$ as $x \to x_0$. That linear map is what we call $f'(x_0)$. The actual mapping $x_0 \mapsto f'(x_0)$ is usually not linear.
Now when you look at $fg$, the situation is essentially the same. You get a linear map $A$ such th... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is $ f(z)= |z|.\bar z $ analytic? I want to know whether it is analytic - and if so, to find $ f´(z)$
What I do:
I use the polar form.
$ z= x + iy \\ |z|=r \\ \bar z= (cos (\phi) + isin(\phi))$
then: $ f(z)= r^2(cos (\phi) + isin(\phi)) = r^2cos (\phi) + i r^2sin(\phi)$
Cauchy-Riemann:
$\begin{cases} \\
\frac{\parti... | Put $z=r.e^{i\theta}$ for polar coordinates then $f(r.e^{i\theta})=r^2(cos\theta-isin\theta)$.
You can see that the real part $u(r,\theta)=r^2.cos\theta$ and imaginary part $v(r,\theta)=-r^2.sin\theta$ satisfy CR-equations
$ru_r=v_\theta, u_\theta=-rv_r$ iff
$3r^2.cos\theta=0$ and $3r^2.sin\theta=0$ i.e. $r=0$
Since C... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that $\mathrm{rank}(A)=\dim(\mathbb{Q}\otimes_{\mathbb{Z}}A)$ Call a subset $X$ of an abelian group $A$ independent if, whenever $\Sigma m_ix_i=0$, where $m_i\in \mathbb{Z} $ and almost all $m_i =0$, then $m_i = 0$ for all $i$. Define $\mathrm{rank}(A)$ to be the number of elements in a maximal independent subset... | A priori, the general element of $\Bbb Q\otimes A$ is a rational linear combination of elements of the form $q\otimes a$ with $q\in\Bbb Q$ and $a\in A$. As $$q\otimes a=q\cdot 1\otimes a$$ and $$\frac nm\cdot 1\otimes a+\frac rs\cdot 1\otimes b=\frac 1{ms}\cdot 1\otimes(nsa+rmb)$$
we can write each element of $\Bbb Q\o... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Show that $\frac{3}{5} + i\frac{4}{5}$ isn't a root of unity Intuitively, I see why $\frac{3}{5} + i\frac{4}{5}$ is not a root of unity because $\frac{2\pi}{\arctan(4/3)}$ appears to be irrational when I plug into my calculator. But how to I show this rigorously?I think contradiction should work, but still I wasnt able... | Hint: An $n$th root of unity has the form $e^{2\pi ik/n}$ with $0\leq k\leq n-1$.
By Euler's formula, $r(\cos \phi + i\sin\phi) = re^{i\phi}$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to solve this Riemann integral problem? Suppose $f$ is Riemann Integrable on $[a,b].$ Is there a real number $c\in [a,b]$ such that
$$\int_a^c f(x)\,dx=\int_c^b f(x)\,dx$$? If it is, then proof it, otherwise, give a counter example.
| If $f$ is Riemann integrable on $[a,b]$ then $f$ is bounded, i.e. there exists an $M$ such that $|f(x)|\leq M$ for all $x\in[a,b]$. This implies that the function
$$F(x):=\int_a^x f(t)\>dt\qquad(a\leq x\leq b)$$
is Lipschitz continuous with constant $M$. Let $\int_a^b f(t)\>dt=:C$. Then $F(a)=0$ and $F(b)=C$. By the in... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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$(\varepsilon, \delta)$ proof for $f(x)=\begin{cases} \frac{3-x}{2} & x<1 \\ x & x \geqslant1 \end{cases} $
Show using $(\varepsilon, \delta)$- definition of continuity that
$$f(x)=\begin{cases}
\frac{3-x}{2} & x<1 \\
x & x \geqslant1
\end{cases}
$$
is continuous at $x=1$.
I had a problem like ... | I would say that you are almost done with the problem, but not quite there yet. What you have is $\delta_\text{left}$ and $\delta_\text{right}$ such that the following holds: if $0<x-1<\delta_\text{right}$ then $|f(x)-f(1)|<\varepsilon$, and if $0<1-x<\delta_\text{left}$ then $|f(x)-f(1)|<\varepsilon$. All that's le... | {
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Acceleration of a ball on a plane I have a plane ($ax+by+cz+d=0$) in a 3D world, and a gravity vector $\vec{g}$ (say it's $[0, 0, -9.81]$.) How would I find the acceleration vector of an object on this plane, ignoring friction?
| What you want to do is project $\vec{g}$ on the plane.
First we will consider a new plane: $ax+by+cz=0$. This plane is parallel to our previous one, so our result won't change, but it goes through the origin.
Then we need a normal vector of this plane. In this case, we're lucky because we have this particular form of... | {
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How to prove that elements in the main diagonal of PD matrix are all positive? Reminder: As PD matrices are defined vector X is not the vector 0
Given a positive definite matrix A which is symmetric
We need to prove that he following elements in the main diagonal are all positive
(A(1,1) A(2,2) ... A(n,n))
I started ... | $A=[a_{i,j}]$ is the matrix of the scalar product $<x,y>=x^TAy$.
Then $<e_i,e_i>=||e_i||^2=e_i^TAe_i=a_{i,i}>0$.
| {
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Level Curves Problem Show that $x^2+y^2=6$ is a level curve of $f(x,y)=\sqrt{x^2+y^2}-x^2-y^2+2$.
I know that the first equation is a circle but I do not know how to find out if the second one is it too.
Thanks for the help.
(Sorry my English is not good).
| With
$x^2 + y^2 = c, \; \text{ a constant}, \tag 1$
for any $c$, we have
$f(x, y) = \sqrt{x^2 + y^2} - x^2 - y^2 + 2 = \sqrt{x^2 + y^2} - (x^2 + y^2) + 2 = \sqrt c - c + 2; \tag 2$
thus the circle (1) lies in the $\sqrt c - c +2$-level set of $f(x, y)$; note we needn't prove
$f(x, y) = c_0, \; \text{a constant} \tag... | {
"language": "en",
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Show that $|\{x\}^2-\{x\}+1/6|\leq \frac{1}{6}$ I am trying to show the following inequality holds for $x>0$:
$$|{x}^2-{x}+1/6|\leq \frac{1}{6}.$.
I was able to show that $\{x\}^2-\{x\}+1/6 \leq 1/6$ because $\{x\}^2-\{x\}<0.$ However I am having trouble with showing $\{x\}^2-\{x\}+1/6\geq -\frac{1}{6}$. This seems to ... | Set $y=\{x\}$. As $x>0$, we have $0\le y <1$, so, as the roots of $y^2-y$ are $0$ and $1$, its minimum is attained at $\frac12$, and we have $-\frac14\le y^2 -y\le 0\:$ on $[0,1)$. Therefore
$$-\frac1{12} \le y^2-y+\frac 16\le \frac16.$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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When is it true that $x^2 < \lfloor{x}\rfloor \lceil{x}\rceil$? When is it true that $x^2 < \lfloor{x}\rfloor \lceil{x}\rceil$? It seems like this should be true whenever $x$ is close to $\lfloor{x}\rfloor$ than $\lceil{x}\rceil$, but I'm not sure how to prove this. I am trying to show that this is equivalent to $\frac... | Hint:
First notice, that when $x$ is an integer, the inequality does not hold.
Let's write $x=n+\alpha$, where $n$ is an integer and $0 < \alpha < 1$, then we can rewrite the inequality as $(n+\alpha)^2 < n(n+1)$. Now the problem is reduced to solving the following inequality:
$$\alpha^2 + 2n \alpha - n < 0$$
Can you ... | {
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Finding the closed-form formula to a recurrence with summation of terms I am currently trying to work through this recurrence problem but am having a hard time coming up with the solution:
$g\left(n\right)=\left(\sum_{i=1}^{n-1}g\left(i\right)g\left(n-i\right)\right)+1$
Where the base $g(0)=0$.
One thing I noticed was ... | We can obtain a generating function for your sequence. First we modify the recurrence relation to state $$g(n) = 1 + \sum_{i=0}^ng(i)g(n-i).$$ Note that this gives the same relation because $g(0)=0$. Also note that this equality does not hold for $n=0$ (that gives $g(0) = g(0)^2 + 1$). Now we define the generating func... | {
"language": "en",
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"source": "stackexchange",
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Combinatorics and integer solutions to equations Here is the question:
How many non-negative integer solutions are there to the equation $x_1 + x_2 + x_3 + x_4 = 74$ with each $x_j \leq 26$?
We've been instructed to use the $C(n+r-1,r-1)$ identity for the amount of integer solutions for the equation $x_1 + x_2 + \cdots... | Let $U$ be the number of solutions where you don't restrict any of the $x_j$.
Now, let $S_j$ be the number of solutions restricting only $x_j$. Define $y_j = x_j - 27$. Then you have the equation for, for $S_1$ and $x_1$ and $y_1$ for instance,
$$(y_1 + 27) + x_2 + x_3 + x_4 = 74$$
Simplify to $y_1 + x_2 + x_3 + x_4 = ... | {
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Find the missing angle in the triangle Given the following triangle $ABC$, find the angle of $ACD$.
Notice that $AC = BC$, I can find the orthocenter of $ABC$. However, I am stucked and do not know what is the next step. Any hint would be appreciated.
| Applying the trigonometric form of the Ceva's theorem we obtain:
$$\frac{\sin\angle ACD}{\sin\angle BCD}=\frac{\sin40^\circ\sin50^\circ}{\sin30^\circ\sin20^\circ}=\frac{2\sin40^\circ\cos40^\circ}{\sin20^\circ}
=\frac{\sin80^\circ}{\sin20^\circ}=\frac{\cos10^\circ}{2\sin10^\circ\cos10^\circ}=\frac{\sin30^\circ}{\sin10^\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3618767",
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Solving $\lfloor|x+1|-|x|\rfloor\geq x^2$ I need to solve this inequality:$$\lfloor|x+1|-|x|\rfloor\geq x^2.$$
I checked the domains:
*
*$x\geq0$ resulted in $[0,1]$.
*$x\leq-1$ resulted in a contradiction.
But when trying to solve the inequality for $-1<x<0$ I get stuck:
$$
\lfloor|x + 1| - |x|\rfloor = \lfloor ... | $\lfloor 2x + 1\rfloor$ takes on two possible values for $-1 < x < 0$: $-1$ and $0$.
It takes value $-1$ if $2x + 1 < 0$, which happens when $x < -\frac12$; in this case, $\lfloor 2x + 1\rfloor$ cannot be larger than $x^2$, since $x^2$ is positive and $-1$ is not.
It takes value $0$ if $0 \leq 2x + 1 < 1$, which happen... | {
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Structure of set invariant under multiplication by a multiplicative subgroup Suppose $p$ is prime. What can be said about the structure and the size of a set $ A \subseteq F^*_p$ ($F^*_p $is a multiplicative group of integers modulo $p $), provided that it does not change under multiplication by a multiplicative subgro... | In general, for any group $U$, let $G$ be a subgroup of $U$ and let $A$ be any subset of $U$. It is easy to check that the condition $GA=A$ is equivalent to: $A$ is a union of right cosets of $G$. This includes "$A$ is a subgroup of $U$ containing $G$" but is broader.
For example, let $p=7$, and let $G=\{1,6\}$ (consid... | {
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Inequality between $\mathbb{E}[XY]$ and $\mathbb{E}[X^2]$ if $X$ and $Y$ have the same distribution Does there exist an inequality connecting $\mathbb{E}[XY]$ and $\mathbb{E}[X^2]$ if $X$ and $Y$ have the same distribution, regardless of whether they are independent or not?
| Note that $0\le E(X-cY)^2=c^2EY^2-2cEXY+EX^2$ for all $c\in\Bbb R$, so this quadratic in $c$ has discriminant $\le0$, i.e. $4c^2(EXY)^2-4c^2EX^2EY^2\le0$. This simplifies to $|EXY|\le EX^2EY^2$.
| {
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"timestamp": "2023-03-29T00:00:00",
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Why is the assumption $(dx)^2 = 0$ actually correct instead of just approximately correct? Imagine dividing a sphere into concentric spherical shells of thickness $dr$ and inner radius $r$.
The volume of each shell is $$dV = \frac{4\pi}{3} [ (r + dr)^3 - r^3]$$
Expand the cubic expressions, we get:
$$
(r + dr)^3 - r^3... | Instead of thinking of $dr$ as a number, think of the manipulations you've used to get $dV=4\pi r^2dr$ as a shortcut for computing $\frac{dV}{dr}$ using the limit definition of the derivative: $\lim_{h\to0}\frac{3r^2h+3rh^2+h^3}h=3r^2$. What you've calculated isn't really the volume of any particular shell, but the lim... | {
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Is there typographical error in Stephen Willard's General Topology proof of Theorem 28.11 Here is 28.11:
The proof on page 206 initially refers to separation order E(a,b). In the second paragraph, it supposes distinct points in E(a,c) - {a,b}. I am reading this on my own so I have no-one else to ask. I believe it shou... | Yes. There is an error. Brian Scott confirmed this in the comments section.
| {
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Maximizing $f(x,y) = (a + x)(b + y)$ under the constraint $d=x+y$, where $a$, $b$, $d$ are known.
Find the maximum of the function $f(x,y) = (a + x)(b + y)$ under the constraint $d = x + y$, where $a$, $b$, $d$ are known.
It seems "obvious" to me that you'd want to split it between the two so that both sides of the p... | It's labeled as pre-calculus problem so I'll solve it without calculus.
$f(x,y)=(a+x)(b+y)=(a+x)(b+d-x)=-x^2+(b+d-a)x+(ab+ad)=-(x-(b+d-a)/2)^2+(ab+ad+(b+d-a)^2/4)$
The maximum can be obtained at $x=(b+d-a)/2$ and the maximum value is $ab+ad+(b+d-a)^2/4$
| {
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Is there a self-map of the disk with Jacobian everywhere greater than $1$? It might be silly, but I am not sure how to approach this problem.
Let $D \subseteq \mathbb{R}^2$ be the closed unit disk. Does there exist a smooth map $f:D \to D$ such that $\det df >1$ everywhere?
I don't assume $f$ maps boundary to boundar... | Choose an $N\gg1$, and map $D$ to an ellipse $E\subset R:=[-2N,2N]\times\bigl[-{1\over N},{1\over N}\bigr]$ by putting
$$f_1:\quad (x,y)\mapsto \bigl( 2Nx,{1\over N}y\bigr)\ .$$
We have ${\rm det}(df_1)=2$. Now use a map $f_2$ with ${\rm det}(df_2)\approx1$ to wrap the long rectangle $R$ essentially area preserving ... | {
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What are the continuous functions $ x f(y)+y f(x)=(x+y) f(x) f(y) ? $ question -
What are the continuous functions on $\mathbb{R}$ which are solutions of the equation
$$
x f(y)+y f(x)=(x+y) f(x) f(y) ?
$$
my try -
by putting $y=x$ i get $f(x)=0$ or $1$ for all $x$ not equal to $0$...
now my answer is same as mention ... | Wit $x=y$, we arrive at the necessary condition
$$2xf(x)=2xf(x)^2 $$
and hence $$\tag1f(x)\in\{0,1\}\quad\text{for }x\ne0. $$
The only continuous functions with this property are the constant functions
$$f(x)=0$$
and
$$f(x)=1. $$
Both are directly verified to indeed solve the original functional equation.
What if we d... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Classical v/s Bayesian Hypothesis Testing This question has 2 parts:
(1) What is the fundamental difference between classical and bayesian hypothesis testing? How do I interpret this difference.
(2) Here is a paragraph quoted from Casella and Berger Statistical Inference (Section 8.2):
I don't understand:
(i) Why is P... | With the coin tossing, the quantity of interest in the distribution is the probability the outcome is heads/tails. This is a fixed and constant number in the classical paradigm. For a fair coin, this probability is $0.5$ irrespective of whether you get an actual heads or tails (i.e. irrespective of the data X) and $P(H... | {
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Remove minimum number of nodes to make graph disconnected Find the minimum number of nodes that need to be removed to make graph disconnected( there exists no path from some node x to all other nodes). Number of nodes can be 105
| You are searching for the minimum $k$ such that your graph $G = (V, E)$ is $k$-vertex-connected.
To solve this problem consider Menger's theorem:
Let $G$ be a finite undirected graph and $x$ and $y$ two nonadjacent vertices. Then the size of the minimum vertex cut for $x$ and $y$ (the minimum number of vertices, disti... | {
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How to calculate the centre of a circle given two points and the equation of a line going through the centre? A(3,5) and b(9,-3) lie on a circle. show that the centre of the circle lies on the line 4y-3x+14 = 0
Is it possible to work out the centre of the circle algebraically since the only method I can think of is dra... | You must rember this teorem:
In a circunference, the center of the circle belongs to the perpendicular line drawn from the middle point of a chord.
So, we must calculate the middle point of $AB$, that is:
$$M(6,1)$$
Then we have to find the line passing throught $A$ and $B$, so:
$$y=-\frac{4}{3}x+15$$
Now, we can cal... | {
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"timestamp": "2023-03-29T00:00:00",
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Gradient vector for a function of two variables For the function $z = x^2 + y^2 , z = f(x,y)$
the gradient comes out to be $(2x,2y)$.
However since the function is in 3D , shouldn't the gradient also be a 3D vector?
How would the gradient change if $w = f(x,y,z) = x^2 + y^2 -z =0$?
Will it be $(2x,2y, -1)$?
| Maybe you confuse $f$ with its graph. The graph of $f$ is three dimensional, i.e., a subset of $\mathbb{R}^3$. But $f$ has only two entries. For every partial differentiable function $f = f(x, y)$ the gradient of $f$ is defined as $(\partial_x f, \partial_y f)$, so the gradient is a planar vector in this case.
If you c... | {
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sufficient condition for mean convergence in $L^1$:prove or disprove Let $f_n \; \colon (0,1) \to \mathbb{R} $ a sequence of Lebesgue integrable functions which converges almost everywhere in $(0,1)$ to $0$. Prove or Disprove: if there exists $p \in (1,+ \infty)$ such that $(f_n)$ is bounded in $L^p(0,1)$,then $\lim_{ ... | I think the property is true and Egorov's theorem will help here. Fix $\varepsilon >0$. Since $f_n \to 0$ a.e., there is a set $E \subset (0,1)$ such that $\lvert E\rvert < \varepsilon$ and $f_n \to 0$ uniformly on $(0,1)\setminus E$. Now fix $\delta > 0$ and $N$ large enough that $\lvert f_n \rvert < \delta$ on $(0,1)... | {
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"question_score": "2",
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"answer_id": 0
} |
The right way to use the chain rule for composite functions like $f(g(x,y),h(x,y))$ Im in doubt about a resolution. I made it in a way that gave me the right answer but I don't think it's the right way to answer. I wish some one can help me understand the right way to make.
So, $f:\Bbb{R}^2\rightarrow\Bbb{R}$ is a deri... | Let’s try a concrete example: suppose we have $f:(x,y)\mapsto x^2+y^2$. Then $$z = f(x-y,y-x) = (x-y)^2+(y-x)^2=2(x-y)^2$$ and ${\partial z\over\partial x}=4(x-y)$, which is not identically zero.
What went wrong with your calculations? You’ve made the common mistake of using the same names to mean different things, w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3621355",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Calculate $\lim_{n\to\infty} \frac{ (1^{1^p}2^{2^p}\cdot...\cdot n^{n^p})^{ 1/n^{p+1} }}{n^{1/(p+1)}}$
Calculate:
$$\lim_{n\to\infty} \frac{ (1^{1^p}2^{2^p}\cdot...\cdot n^{n^p})^{ 1/n^{p+1} }}{n^{1/(p+1)}}$$
I've done some steps as follows: $$a_n:=\frac{ (1^{1^p}2^{2^p}\cdot...\cdot n^{n^p})^{ 1/n^{p+1} }}{n^{1/(p... | I seem to remember answering this question sometimes, but I didn't find it! So, I write the answer again, I didn't COPY my previous answer. Thank @metamorphy for pointing out this！The following is my previous answer.
Computing limit of a product
$$\frac{1}{n}\sum_{k=1}^n\big[\big(\frac{k}{n}\big)^p\ln \frac{k}{n}\bi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3621545",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
Getting the differential equation back from its solutions.
Find a linear differential equation with constant coefficients satisfied by all the given
functions.$u_1(x) = x^2$, $u_2(x) = e^x$, $u_3(x) = xe^x$.
How do I proceed with this? I know that 1 is a twice repeated root of characteristic equation, but what can ... |
''I know that $1$ is a twice repeated root of characteristic equation''
So you know that $k=1$ solves the characteristic equation $ak^2 + bk + c =0$. Then, you could have, for example $a = 1, b = -2, c = 1$. Since the characteristic equation corresponds directly to the homogeneous linear SODE, we have
\begin{equation... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3621741",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Find the minimum value of $f$
Find the minimum value of $$f(x)=\frac{\tan \left(x+\frac{\pi}{6}\right)}{\tan x}, \qquad x\in \left(0,\frac{\pi}{3}\right).$$
My approach is as follows. I tried to solve it by segregating it
$$f(x)=\frac{1}{\sqrt{3}\tan x}+\left({\sqrt{3}}+\frac{1}{\sqrt{3}}\right)\frac{1}{\sqrt{3}-\t... | We know that $\tan(x)\tan(y)=1$ then $x+y=\frac{\pi}{2}$ thus we have $f(x)=\frac{1}{\tan(x)\tan(\frac{\pi}{3}-x)}$ thus we need to maximize denominator (lll be referring to it as k)to get minimum value . Now using the fact that $\tan(x+y)=\frac{\tan(x)+\tan(y)}{1-\tan(x)\tan(y)}$ we have $\tan(\frac{\pi}{3})=\frac{\ta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3621911",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 2
} |
Trouble to understand an Analysis proof. I am currently studying the following proof, which may be found on this article (page 12):
However, I'm facing difficulties to proper understand some of the steps. Here are my questions:
*
*Why is it possible to say that "there exists an integer $N$ and $\alpha > 0$ such tha... | (1) $x < x_1$, so by defn of inf, $\limsup_{n \to \infty} \frac{\log\Sigma_n(x)}{\log n} < 1$, call the limsup $1-2\alpha$ for $\alpha > 0$. Then there is some $N$ so that for all $n \ge N$, $\frac{\log \Sigma_n(x)}{\log n} \le (1-2\alpha)+\alpha = 1-\alpha$. In other words, there is some $N$ so that for all $n \ge N$,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3622040",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Let $\alpha(f)=|f(0)|^2+\int_0^1f(x)^2dx$,find the maximum value of $\alpha(fg)/\alpha(g)$ Let $\alpha(f)=|f(0)|^2+\int_0^1|f(x)|^2dx$ be a functional on $C^2[0,1]$, the space of real functions o $[0,1]$ whose 2-oder derivative is continuous. Determine $\textrm{sup}\{\alpha(fg)/\alpha(g): g\not=0\}$, where $f$ is fixed... | The sumpremum is equal to $$\max\{|f(t)|^2:\ t\in[0,1]\},$$ i.e., the square of the infinity norm of $f$.
We need to maximize $\alpha(fg)$ for $g$ with $\alpha(g)=1$. So
$$
\alpha(fg)=|f(0)|^2\,|g(0)|^2+\int_0^1|f(t)|^2\,(g(t)|^2\,dt\leq\|f\|_{\infty}^2.
$$
Now if, $\|f\|_\infty=|f(0)|$, choose a twice-differentiable ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3622176",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Understanding the Fundamental theorem of Calculus in plain english I am learning Calculus. I am trying to understand the fundamental theorem of calculus. I am following this wikipedia article: https://en.wikipedia.org/wiki/Integral.
I am having a hard time understanding what they refer to as the the Fundamental theorem... | The FTC says that integration and differentiation are inverse operations. If you differentiate the right kind of integral, then you get the integrand back. If you integrate a derivative, you get the original function back.
D(I(f)) = f
I(D(f)) = f.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3622287",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 5
} |
Question about $f(x)=\sum_{k=1}^\infty (-1)^{k+1}\sin (\frac{x}{k}) $ This function is rather peculiar. It is easy to establish the following:
$$f(x) =\sum_{k=0}^\infty (-1)^k A_{2k+1} \cdot x^{2k+1}, \mbox{ with } A_k=\Big(1-\frac{1}{2^{k}} + \frac{1}{3^{k}}- \frac{1}{4^{k}}+\cdots\Big).$$
Note that $A(1)=\log 2$, and... | Distribution of Roots - Example
There was a question about the distribution of roots and extrema values of this function. From the analysis above, it sufficient to study first $4K^{\prime}$ elements of the series, as it should be a good approximation.
It is intuitive to group values of $x$ having the same integral part... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3622469",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 1
} |
Maximum number of iterations of a simple algorithm Suppose there is a 0-1 string of length n. We can perform the following operation on the string:
We can choose two zeros and invert the subsequence between them. The inversion includes the two zeros aswell. For example if the string is 011010, and we choose the first a... | Let $n$ be the length of the word. For even $n$, a working idea is to build a metric for the word so that every transition increases the metric, and so that the strategy you proposed increases the metric in every step exactly by $1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3622598",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Decomposing a function's variable and adding up the partials of the parts equals the original partial? Theorem
Let $f(x)$ and $g(x_1, x_2, \ldots, x_n)$ be differentiable and equal when
$x_1 = x_2 = \ldots = x_n = x$. Then
$$\frac{\partial f}{\partial x} = \frac{\partial g}{\partial x_1} + \frac{\partial g}{\partial... | Define $\mathbf{X}(x)=(x,...,x)$, now we have that $f(x) =g(\mathbf{X}(x))$, try taking an $x$ derivative of both sides making sure to use the multivariable chain rule and you'll obtain your theorem!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3622741",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
a PDE problem involving divergence
Consider the PDE on a bounded smooth domain $\Omega \subset R^n$, $\triangle u(x)= 0$ on $\Omega$, $\frac{\partial u(x)}{\partial n}|_{\partial \Omega}=g(x)$. Prove that if it admits a smooth solution u, we have $\int_{\partial \Omega} g d \sigma=0$
I know that $\triangle u=div(\tri... | We have
$\nabla \cdot \nabla u = \triangle u = 0 \tag 1$
on $\Omega$. Thus the divergence theorem yields
$\displaystyle \int_{\partial \Omega} \nabla u \cdot \mathbf n \; dS = \int_\Omega \nabla \cdot \nabla u; dV = \int_\Omega 0 \; dV = 0, \tag 2$
where $\mathbf n$ is the outward-pointing unit normal vector field on ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3622912",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Are there geodesic triangles on surfaces with non-constant curvature with angle sum 180? I've been reading about differential geometry and the Gauss-Bonnet theorem to write a paper for my geometry class and am interested specifically in geodesic triangles on surfaces.
I was wondering if it is possible to create a geod... | Just consider your favorite flat triangle T in a euclidian plane. the consider a $C^\infty$ fonction $f: T\to \bf R$ wih is $0$ in the neighbourhood of the boundary of $T$. Now the triangle which is the graph of this fonction ha $>0$ curvature near the point where $f$ is maximal or minimal. It cannot be everywhere posi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3623080",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Evaluate $\lim_{x\to 1}\frac{x^{x^a}-x^{x^b}}{\ln^2 x}$
Evaluate $$\lim_{x\to 1}\frac{x^{x^a}-x^{x^b}}{\ln^2 x}$$
I've tried to use fundamental limits to solve it and conclude that the limit is $a-b$ as follows:
\begin{align*}
\lim_{x\to 1}\frac{e^{x^a\ln x}-e^{x^b\ln x}}{\ln^2 x}&=\lim_{x\to 1}\frac{e^{x^a\ln x}-1-(... | It's not clear to me what happens in the step from blue to red:
$$\begin{align*}
\lim_{x\to 1}\frac{e^{x^a\ln x}-e^{x^b\ln x}}{\ln^2 x}&=\lim_{x\to 1}\frac{e^{x^a\ln x}-1-(e^{x^b\ln x}-1)}{\ln^2 x}\\
&=\color{blue}{\lim_{x\to 1}\left(\frac{e^{x^a\ln x}-1}{\ln^2 x}-\frac{e^{x^b\ln x}-1}{\ln^2 x}\right)}\\
& = \color{re... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3623286",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
f(x) increasing or decreasing
Let $f(x) = x + 2x^2\sin(1/x)$, with $x\ne0$,
and $f(x)= 0$ when $x= 0$. Detrmine if $f(x)$ is increasing or decreasing at $x= 0$
my attempt: using first principle it is easy to see that $f'(0)=1$ however if we find derivative of $f(x)$ as $f'(x) = 1+4x\sin(1/x) -2\cos(1/x)$ ,as $f'(... | $$f(0)=0$$ and
$$f(\epsilon)=\epsilon+2\epsilon^2\sin\frac1{\epsilon}=\epsilon\left(1+2\epsilon\sin\frac1\epsilon\right)>f(0)$$ for all $\epsilon<\frac12$. Similarly $f(-\epsilon)<f(0)$ and the function is indeed growing at $x=0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3623433",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Is there a functor $F$ of left modules preserving $\oplus$ by arbitrary isomorphism, but its restriction to fin. gen. proj. modules isn't additive? Are there rings $R$, $S$ and a functor $F:{_R\textbf{Mod}}\to{_S\textbf{Mod}}$ such that
*
*For all left $R$-modules $M,N$, we have $F(M\oplus N)\cong F(M)\oplus F(N)$
v... | Let $R$ be a field $k$, $S=\operatorname{End}_k(k^{\oplus\mathbb{N}})$, and $F(M)=S\otimes_k (M\otimes_k M)$. Note that $S$ has the property that $S\cong S^n$ as an $S$-module for all finite $n>0$. If $M$ is nontrivial and finite-dimensional, then so is $M\otimes_k M$, so $F(M)\cong S$. If $M$ is infinite-dimensiona... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3623570",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A structure which looks almost like a semi-ring. Today I have encountered an interesting structure, similar to that of a ring or a semi-ring.
It is a structure $(S, +, \cdot, 1)$, where $S$ is a set, $+, \cdot$ are binary operations, and $1\in S$.
$(S, \cdot, 1)$ is a commutative monoid, $(S, +)$ is a commutative semig... | Hopefully, I am not saying anything stupid here.
Consider such a set $S$. Define $R= S \cup \{ 0_R \}$ with the operations extended by
$$
0_R+x =x \\
0_R\cdot x= 0_R$$
Then $R$ becomes a commutative semi-ring without zero divisors (i.e. $xy=0$ implies $x=0$ or $y=0$).
Converesely, let $R$ be any commutative semi-ring ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3623902",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to prove a formula for Gamma function I made some observation for Gamma function
Suppose
$$x=a+i b,a\in \mathbb{R},b\in \mathbb{R}$$
Then
$$
\left| \cos \left(\frac{\pi (a+i b)}{2}\right) \Gamma (a+i b)\right|\to\left| \sqrt{\frac{\pi }{2}} (a+i b)^{a-\frac{1}{2}}\right|
$$
When
$$
a\in [0,1],b\to\infty
$$
How can ... | Let
$$
x=a +i b,a \in \mathbb{R},b\in \mathbb{R}
$$
Use following formula from
Bateman, Harry (1953) Higher Transcendental Functions, Volumes I, p.47, (6)
$$
\left| \Gamma (x)\right| \to \left| \sqrt{2 \pi } e^{-\frac{1}{2} (\pi b)} x^{a -\frac{1}{2}}\right|,a \in [0,1]
$$
Expand cosine
$$
\left| \cos \left(\frac{\pi ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3624042",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Evaluating $\int_{-5}^{\sqrt{x}}(\frac{\cos t}{t^{10}})dt$
Evaluate $y=\int_{-5}^{\sqrt{x}}(\frac{\cos t}{t^{10}})dt$
I've tried differentiating both sides of the fraction until the denominator was 1, and then integrating that by parts, but this was marked wrong. I know I can't just integrate by parts right off the ... | First note that this integral is improper because $-5 < 0<\sqrt x.$ Split off a bad piece near 0 by writing $c=\min(5,\sqrt x,\pi/3)$ so $$\int_{-5}^\sqrt x \frac{\cos t}{t^{10}}dt=\int_{-5}^{-c} \frac{\cos t}{t^{10}}dt+\int_{-c}^c \frac{\cos t}{t^{10}}dt+\int_c^\sqrt x \frac{\cos t}{t^{10}}dt$$. Of the three integrals... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3624116",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 3
} |
Finding det(A) with standard basis vectors For example, let $e_1 = [1 \quad 0]$ and $e_2 = [0 \quad 1] $be standard basis vectors. A is a $2 \times 2$ matrix. $Ae_1 = [-3\quad 7]$ and $Ae2 = [3 \quad 5] $
How do I find the $\det(A)$?
| Since A is a 2x2 matrix, simply use the formula for finding determinants for 2x2 matrices.
\begin{bmatrix}a&b\\c&d\end{bmatrix}
The formula is ad-bc. In this case, A$e_1$ = \begin{bmatrix}-3\\7\end{bmatrix}
And A$e_2$ = \begin{bmatrix}3\\5\end{bmatrix}
Combine these vectors to get: \begin{bmatrix}-3&3\\7&5\end{bmatrix... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3624245",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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About the theorem of dual basis Theorem for dual basis: Let V be a finite dimensional vector space and $\beta={u_1,u_2,...,u_n}$ be an ordered basis for V. Then there exists a basis $\beta^*={f_1,...,f_n}$ of $V^*$ such that $f_i(u_j)=\delta_{ij}$
So what is $f_i(u_j)=\delta_{ij}$ in terms of significance in a linear f... | In all of the contexts I've seen it being used, you have
$$\delta_{ij} = \begin{cases}
1, & \text{ for } i = j \\
0, & \text{ for } i \neq j
\end{cases}\tag{1}\label{eq1A}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3624386",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Hilbert's Hotel Paradox: Guests moving to new room every day? Suppose there are infinitely many coaches with infinitely many members in each coach. They stay at the hotel for infinitely many days. I know that guests can be accommodated using various methods like the prime powers method, but there's a slight variation i... | A far simpler approach than prime powers is as follows. Number the rooms starting from $0$. Each day,
*
*guests in even-numbered rooms move two rooms up
*guests in odd-numbered rooms move two rooms down, except for the one in room $1$, who moves to room $0$
This creates an infinite cycle linking every room, on whic... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3624538",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 2
} |
Combinations problem - Finding the number I would like to ask for your confirmation to my thought in regards with the following exercise:
One man has 30 different statues, 27 genuine and 3 fake. He sold 10 of these statues in museum A, 10 in museum B and 10 in museum C. What is the probability that each of these museum... | Your answer seems wrong. Although, from your answer I am unable to judge about how you exactly arrived at it but here's a hint to how you could have approached the problem:
Just in case you don't know: Number of ways to divide n objects in n1 groups of m1 object, n2 groups of m2 object and so on till nk groups of mk ob... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3624715",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Cramer-Rao bound for LS estimator It's a problem from Machine Learning: A Bayesian And Optimization Perspective (problem 3.7):
Derive the Cramer-Rao bound for the LS estimator, when the training data result from the linear model $$y_n=\theta x_n+\eta_n, n=1, 2, ..., N$$ where $x_n$ and $\eta_n$ are i.i.d sample of a ze... | This is partly a summary, and thanks to works by @a_student, @StubbornAtom, and @jld in https://stats.stackexchange.com/q/320600. This may be the final answer to this problem, please point out mistakes if you find some.
First, we compute the C-R bound of the estimator, by the definition of
$$
I_{(X, Y)}(\theta)=-\math... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3624887",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Likely fake proof of the irrationality of a rational multiple of pi/pi I would like to know what has gone wrong in this 'proof'.
Suppose that $$k\frac 1\pi\pi=\frac ab\operatorname{,where}a,b\in \mathbb Z\operatorname {and}k\in \mathbb Q.$$
Then, we multiply both sides by $i$ and raise $e$ to the power of both sides.... | The main problem is that some of the rules for working with powers from real numbers are no longer true with complex numbers.
While for 2 complex number $z_1,z_2$ the rule $e^{z_1+z_2}=e^{z_1}e^{z_2}$ is still correct, the version dealing with multiplied exponents isn't, generally we have
$$e^{z_1z_2} \color{red}\neq ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3625057",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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Proof equation $\lim_{n \to \infty}^{}\sum_{k=1}^{n}k^{p}/(n+1)^{p}= \frac{1}{p+1}$ Just tried some approach (Stolz-Cesàro theorem and sandwich theorem)
can't prove this equation
$$\lim_{n \to \infty}^{}\frac{\sum\limits_{k=1}^{n}k^{p}}{(n+1)^{p}} =\frac{1}{p+1}$$
| [Assuming the correction in Eeyore Ho's answer is correct]
The easiest way must be by using the bounds: $$\int_1^n x^p\; dx\leq \sum_{k=1}^n k^p\leq \int_1^{n+1} x^p\; dx$$
However, you can also avoid using calculus. Instead it can be proven by induction.
We know that each sum of $p$th powers up till $n$ equals some... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3625242",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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If pair of tangents to a circle in the first quadrant is $6x^2-5xy+y^2=0$ and if one point of contact is $(1,2)$, find the radius. The tangents are $2x-y=0$ and $3x-y=0$. Let the radius be $r$ and centre be $(h,k)$
$$r=\frac{|3h-k|}{\sqrt {10}}$$
$$r=\frac{|2h-k|}{\sqrt 5}$$
$$(h-1)^2+(k-2)^2=r^2$$
I invested a conside... | The perpendicular to $y=2x$ throught $(1,2)$ is:
$$y=-\frac{1}{2}x+\frac{5}{2}$$
The line bisector of the two lines $y=2x$ and $y=3x$ is:
$$y=(\sqrt2+1)x$$
The other bisector line is:
$$y=(\sqrt2-1)x$$
but in this case the circunference wouldn't be tangent to either $y=2x$ and $y=3x$ lines.
Now, we have to inresect the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3625404",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why $a$ is not an essential singularity and how $f$ can be extended in our online lecture we had the following statement but I don't see exactly why it's true , if anyone could explain me why it would be appreciated.
We have $a \in \mathbb{C}$ a complex number and $r \in \mathbb{R}$ with $r>0$. Let $U=\mathbb{D}(a,... | The singularity of $f$ at $a$ is either a pole, a removable singularity or an essential singularity.
If it were a pole of order $n$, you'd have $f(z) = c (z-a)^n + O((z-a)^{n-1})$ for some $c \ne 0$. By having $z$ approach $a$ from a direction such that $c (z-a)^n$ is on the negative real axis you'd get $\text{Re}(f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3625544",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why does stability of $\varphi(x,y)$ imply that for the Shelah local-2-rank, $R_\varphi(x=x)$ is finite . I was reading Artem Chernikov's "Lecture notes on stability theory".
He defines Shelah's local-2-rank $R_{\Delta}$ (taking values in $\mathbb{N}\cup\{\pm \infty\}$) recursively. By definition, $R_\Delta(p) \ge 0$ i... | We can express the properties of the tree we want with a set $\Sigma$ of first-order formulas. Introduce a constant symbol $b_\sigma$ for every $\sigma \in 2^{<\omega}$. Then for every branch $\eta \in {}^2 \omega$ we can add formulas to $\Sigma$ saying that every finite part of that branch is consistent. That is, for ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3625676",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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} |
Integral by parts $\int_0^\infty e^{-st}\frac{\sin(t)}{t} dt $ I want to solve the following integral by parts:
$$\int_0^\infty e^{-st}\frac{\sin(t)}{t} dt $$
I have been trying but I don't know what else to do. The result should be $\frac{\pi}{2}-\arctan\left(s\right) $. I took $\frac{\sin(t)}{t}$ as u and $ e^{-st} $... | Per double integral
$$\begin{align}
\int_0^\infty \frac{e^{-st}}t \sin t \>dt
=\int_0^\infty \int_s^\infty e^{-xt}\sin t \>dx \>dt
= \int_s^\infty \frac1{1+x^2}dx = \tan^{-1}\frac1s\\
\end{align}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3625889",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Showing uniqueness property over a vector space that is the direct sum of two of its subspaces. Let $V$ be a vector space over a field $F$, and let $S$ and $T$ be subspaces of $V$ such that
$V=S⊕T$.
Show that for every $x∈V$ ,there are unique $y_1∈S$ and $z_1∈T$ such that $x=y_1+z_1$. In other words, show that, if $y_... | If $x = a+b, y=c+d$ are the unique decompositions for x and y respectively, then
*
*$P(x+y) = P((a+c)+(b+d)) = a+c = P(x) + P(y).$
*$P(\alpha x) = P(\alpha a+ \alpha b) = \alpha a.$
*$P(P(a+b)) = P(a) = a.$
*$S \subseteq Range(P)$ since $a = a+0$ in $V$.
*$P(a+b) = 0 \Leftrightarrow$ $a=0$. Therefore the kernel... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3625983",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Could $\int\frac{1}{x^{N+1}(x-1)}dx$ be solved analytically? I am trying to solve this integral:
$$\int\frac{1}{x^{N+1}(x-1)}dx$$
I have tried integration by partial fraction, substitution and by parts. But, I can't solve it. So, I would like to ask could this be solved?
Also, May I know when partial fraction does no... | Hint:
Assuming that $N$ is natural, you may do the following
$$\int\frac{1}{x^{N+1}(x-1)}dx = \int\frac{1-x^{N+1}+x^{N+1}}{x^{N+1}(x-1)}dx$$
$$= -\int \frac 1{x^{N+1}}\sum_{n=0}^Nx^n \; dx + \int \frac{dx}{x-1}$$
The first sum comes from
$$\frac 1{x^{N+1}}\frac{1-x^{N+1}}{x-1}=-\frac 1{x^{N+1}}\frac{x^{N+1}-1}{x-1}=-\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3626129",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Evaluating the ratio of beta functions I came across a question which asks for the value of $\alpha$ in the result to the ratio of two $\mathrm{B}$ functions:
$$\frac{\mathrm{B}(m, \frac{1}{2})}{\mathrm{B}(m, m)}=2^{\alpha}$$
I know the results for integer values of $m$, but the question demands that $m>0$ be any real ... | For any value of $m$
$$\frac{{B}(m, \frac{1}{2})}{{B}(m, m)}=2^{2m-1}$$
If you use the gamma function
$$B(m,n)=\frac{\Gamma (m) \Gamma (n)}{\Gamma (m+n)}$$
$$\frac{{B}(m, \frac{1}{2})}{{B}(m, m)}=\frac {\sqrt{\pi }\frac{ \Gamma (m)}{\Gamma \left(m+\frac{1}{2}\right)} } {\frac{\Gamma (m)^2}{\Gamma (2 m)} }=\sqrt{\pi }\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3626271",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Shortest distance from circle to a line
Let $C$ be a circle with center $(2, 1)$ and radius $2$. Find the shortest distance from the line $3y=4x+20$.
This should be very simple, but I seem to end up with no real solutions.
The shortest distance would be from the center of the circle perpendicular to the line right?
S... | Hint. You have shown that your line doesn't intersect the circle. Therefore the shortest distance between the circle and the line is given by the distance between that line and a line such that $$y=\frac43x+c,$$ where the last line is tangential to the circle -- in other words it intersects the circle in just one point... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3626410",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 7,
"answer_id": 5
} |
Are field embeddings unique? Sorry if this is a simple question, as I'm not well versed in field theory.
Suppose a field $K$ has an embedding into $\mathbb R$: $f:K\hookrightarrow\mathbb R$. Is $f$ unique? And if $\mathbb R$ is replaced by an arbitrary field $F$, is the answer still the same?
| $f$ is not unique in general. For instance, there are two embeddings of $\mathbb Q[x]/(x^2-2)$ into $\mathbb R$.
In general, if $K=\mathbb F(\alpha)$, where $\mathbb F$ is the prime field of $K$, and $F$ is a field with the same prime field, then then number of $\mathbb F$-embeddings of $K$ into $F$ is the number of ro... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3626745",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Every square matrix is a sum of two diagonalisable matrices I've been stuck with this question for quite a while and am not sure where to start:
Prove that if $A$ is an $n \times n$ matrix, then $A$ can be written as $B + C$ where both $B$ and $C$ have $n$ distinct eigenvalues. (Hence every square matrix is a sum of t... | You're almost there. Assuming you're working with real matrices, let us denote the diagonal entries as $d_1, d_2, \cdots, d_n$. All you have to do is "split" each $d_i$ into the sum of two numbers $u_i + t_i$, and ensure that all of the $u_i$ and $t_i$ are all different. This is always possible, since for any fixed rea... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3626913",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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I need help understanding the use of | in a specific context. I am reading the wikipedia article on Multiple Sequence Alignments and came across some notation I haven't seen yet, specifically $x_i|i = 1,...,r$ in the statement $L > max \{x_i|i = 1,...,r \}$. I was wondering if anyone can shed light onto what that means... | Given the context of your question, it would mean 'such that'; 'so that'; 'everywhere'.
Or in other words: 'L is greater than the maximum of the set $x_1, x_2, ...x_r$'
In other contexts, it can mean various other things. I suggest you take a good look at:
https://en.wikipedia.org/wiki/List_of_mathematical_symbols
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3627133",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Using the Divergence Theorem on the surface of a unit sphere
Using the Divergence Theorem, evaluate $\int_S F\cdot dS$ , where $F=(3xy^2 , 3yx^2 , z^3)$, where $S$ is the surface of the unit sphere.
My Attempt
$$ \text{div} F = \left(\frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial ... | It is correct, in fact, it is even more simple because you could have directly substituted $\rho^{2}=x^{2}+y^{2}+z^{2}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3627276",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Applying propagator to Laplace eigenfunctions Let $\Omega \subseteq \mathbb{R}^n$ be a non-empty domain with piece-wise smooth boundary and consider a Dirichlet eigenfunction $\varphi$ on $\Omega$. That is, $\varphi : \Omega \to \mathbb{C}$ is a non-trivial solution to $\Delta \varphi = \lambda \varphi$ for some $\lamb... | My answer will be related to the link. I'll provide the necessary context.
Here, $\Delta_g u_j=\lambda_j^2 u_j,$ $p=\frac{1}{2}\left(|\xi|_g-1\right)+\mathcal{O}(h),$ $P:=\text{Op}_h(p)=\frac{1}{2}\left(h^2\Delta_g^2-1\right),$ and $U(t;h)=\exp (-itP/h).$ Note that if we let $h_j=1/\lambda_j,$ then $\text{Op}_{h_j}(p)u... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3627531",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
$I-cP$ Invertible Matrix Question: Assume P is a nonzero $n$ x $n$ matrix, $n \ge 2$, such that $P^2=P$. Let $c\in\mathbb R, c\ne1$. Show that the matrix $I-cP$ is invertible and find its inverse.
I'm having trouble going about this question. By manipulating $Av=\lambda v$, I get that P has eigenvalues $0$ and $1$. I k... | If $c = 0$ then $I-cP$ is clearly invertible. So assume $c \neq 0$.
If $I-cP$ is not invertible, then there is a vector $x \neq 0$ with $(I-cP)x =0$, that is, $x = cPx$. This implies $Px \neq 0$. Multiplying both sides by $P$ we have $Px = cPx$, i.e., $(1-c)Px = 0$, since $Px \neq 0$ we must have $1-c=0$, i.e., $c=1$.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3627705",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
possibility of choosing three from a pool and choosing three from a different pool At the local zoo, a new exhibit consisting of 3 different species of birds and 3 different species of reptiles is to be formed from a pool of 8 bird species and 6 reptile species. How many
exhibits are possible if
a. there are no additio... | It does not matter which order the species are selected. Thus we will (usually) need to divide by $3!$ for the count of ways of selecting the bird and reptile species, something which you didn't do for all the questions.
You also added the counts for the birds and reptiles, which is wrong (it would be true if you were ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3627816",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Finding a polynomial whose roots are connected to the roots of a different polynomial Suppose we have a polynomial function $$f(x) =x^5-4x^4+3x^3-2x^2+5x+1$$ Function $f$ will have 5 roots which can be denoted by $a, b, c, d, e$. I was interested in trying to find a degree 10 polynomial whose roots are given by $abc, a... | Let
*
*$g(x) = x^5 f\left(\frac1x\right) = x^5+5x^4-2x^3+3x^2-4x+1$.
*$S = \{ a,b,c,d,e \}$ be the roots of $f(x)$.
*$T = \{ \frac1a, \frac1b, \frac1c, \frac1d, \frac1d \}$ be the roots of $g(x)$.
*For $I \subset S$ and $J \subset T$, let $\lambda_I = \prod_{\lambda \in I}\lambda$ and $\mu_J = \prod_{\mu \in J}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3628123",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Solving $e^z = 1 $ in complex plane I am solving the equation $e^z = 1 $ in $\mathbb{C}$. The book says, other than $z = 0$, $z = 2 \pi k i$ for $ k \in \mathbb{Z}$ is also the solution. It explains the solution by saying that $e^z$ is periodic function so that
$1 = e^z = e^{2\pi k i}$
However I want to know how the id... | Let $z=x+iy$ where $x, y$ are real. Then $e^z = 2$ means
$$
e^x\cos y + i e^x\sin y = 2,
\\
e^x\cos y = 2\quad\text{and}\quad e^x\sin y = 0
$$
Now $e^x \ne 0$ for all $x$, so from $e^x\sin y = 0$ we get $\sin y = 0$, and thus
$y = n\pi$ for some $ n \in \mathbb Z$. From this we get $\cos y = \cos(n\pi) = (-1)^n$. Bu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3628277",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Complex Analysis. How to use cauchy intergral Given the following integral of gamma on the path $[0,2]$ and $[-2,2]$ we have the integral $$ \int_\gamma \frac{z} {(z^2-1)(z-3)}dz$$
I set it up like
$$ \int_\gamma \frac{\frac{z}{(z+1)(z-3)}}{z-1}=2\pi i $$
and i get that =
$$ \frac{-\pi i}{2} $$
and then i did the fol... | If I understand your notation, by $[-2,2]$ you mean the circle of radius $2$ centered at $-2$. But the integrand's only pole is then at $z=-1$.
So you get $\oint_Cf(z)/(z+1)$, where $f(z)=z/((z-1)(z-3))$. So the integral is equal to $2\pi if(-1)=2\pi i(-1/8)=-\pi i/4$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3628416",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
linear least squares -- complex observations, real estimate constraint Consider the following least squares optimization problem:
$$
\hat{x} = \arg\min_x \| y - A x\|^2
$$
where the observations are complex $y\in{\cal C}^{N\times 1}$, and the complex design matrix $A\in {\cal C}^{N\times K}$ is full rank ($K$). Is the... | Let
\begin{align}
f(x) := \|y - Ax\|_2^2 := \left( y - Ax \right)^*: \left( y - Ax \right),
\end{align}
where $()^*$ is complex conjugate.
Now, let us compute the gradient of $f(x)$ (by computing the differential first), i.e.,
\begin{align}
df(x)
&= \left[ -A^*dx: \left( y - Ax \right) \right] + \left[ \left( y - Ax ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3628616",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Does there exist a sequence $\{a_n\}_{n \ge 0}$ of nonnegative reals such that $ \sum_{j \ge 0} a_{nj} = \frac{1}{n}$ holds for all naturals $n$? Does there exist a sequence $\{a_n\}_{n \ge 0}$ of nonnegative reals such that
$$ \sum_{j \ge 0} a_{nj} = \dfrac{1}{n}$$
holds for all naturals $n$?
My progress: I could show... | Let's put
$$
f(z) = \sum\limits_{0\, \le \,j} {\,a_{\,j} z^{\,j} }
$$
Then the application of the Series Multisection tells us that
$$
\sum\limits_{0\, \le \,j} {\,a_{\,n\,j} z^{\,n\,j} } = {1 \over n}\sum\limits_{0\, \le \,k\, \le \,n - 1} {f(\omega ^{\,k} z)}
$$
Therefore
$$
\sum\limits_{0\, \le \,j} {\,a_{\,n\,j... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3628828",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
Solution verification: $(2x\ln y)\mathrm{d}x+(\frac{x^2}{y}+3y^2)\mathrm{d}y=0$ I've come across this problem while solving an exercise on Exact Differential equations and it's as follows:
Solve the differential equation:
$$(2x\ln y)\mathrm{d}x+\left(\dfrac{x^2}{y}+3y^2\right)\mathrm{d}y=0$$
So what I've noticed i... | $$(2x\ln y)\mathrm{d}x+ \dfrac{x^2}{y}\mathrm{d}y+3y^2\mathrm{d}y=0$$
$$\ln y{d}x^2+{x^2}d \ln y+3y^2\mathrm{d}y=0$$
$$d x^2\ln y+3y^2dy=0$$
Integration gives:
$$x^2\ln y +y^3=C$$
Your answer is perfectly correct.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3628968",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
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