Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Finding asymptotic relationship between: $\frac {\log n}{\log\log n} \overset{?} = (\log (n-\log n))$
Given $f(n)=\frac {\log n}{\log\log n} , g(n)= (\log (n-\log n))$, what is the relationship between them $f(n)=K (g(n))$ where "K" could be $\Omega,\Theta,O$
I thought of taking a log to both sides and see what we ge... | Look at $g(n)$:
$$
g(n) = \log(n-\log n) = \log n + \log\left(1-\frac{\log n}{n}\right) = \log n + o(1)
$$
using the fact that $\frac{\log n}{n} \xrightarrow[n\to\infty]{}0$ and $\frac{\ln(1+x)}{x} \xrightarrow[x\to 0]{}1$.
So $g(n) = \Theta(\log n)$. But $f(n) = \frac{\log n}{\log\log n} = o(\log n)$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Easy way to compute logarithms without a calculator? I would need to be able to compute logarithms without using a calculator, just on paper. The result should be a fraction so it is the most accurate. For example I have seen this in math class calculated by one of my class mates without the help of a calculator.
$... | To evaluate $\log_8 128$, let
$$\log_8 128 = x$$
Then by definition of the logarithm,
$$8^x = 128$$
Since $8 = 2^3$ and $128 = 2^7$, we obtain
\begin{align*}
(2^3)^x & = 2^7\\
2^{3x} & = 2^7
\end{align*}
If two exponentials with the same base are equal, then their exponents must be equal. Hence,
\begin{align*}
3x & ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1653374",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "41",
"answer_count": 9,
"answer_id": 2
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Matrix rank and number of linearly independent rows I wanted to check if I understand this correctly, or maybe it can be explained in a simpler way: why is matrix rank equal to the number of linearly independent rows?
The simplest proof I can come up with is: matrix rank is the number of vectors of the basis of vector ... | Two facts about elementary row operations are useful to resolve this question:
*
*Elementary row operations alter the column space but do not alter the linear dependences among the columns. For example, if column 10 is $4$ times column 3 minus $7$ times column 5, then after doing any elementary row operation, column... | {
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What are the invariants of a number field? How is defined an invariant of a field? Given a certain field extension $L/K$, is it related with the Galois group ${\rm Gal}(L/K)$?
In the case of number fields, which are the invariants associated to these extensions? For example, I remember that the relative discriminant i... | I will try it with a short answer (to the title question), only giving two further invariants. Besides the discriminant also the ideal class group and its order, the class number
are invariants, and the ring of integers of a number field in general. Furthermore
Minkowski's bound, and the invariants involved there, i.e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1653686",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Possible ways to have $n$ bounded natural numbers with a fixed sum Is it possible to count in an easy way the solutions of the equations and inequalities $x_1+x_2+\cdots+x_n = S$ and $x_i\leq c_i$ if all $x_i$ and $c_i$ are natural numbers?
| We know that the number of non-negative integral solutions to the system:
$\begin{cases} x_1+x_2+x_3+\dots+x_n = S\\
0\leq x_1\\
0\leq x_2\\
\vdots\\
0\leq x_n\end{cases}$
is $\binom{S+n-1}{n-1}$, seen by stars-and-bars counting. In the event that you do not consider zero to be a natural number, then it is as though y... | {
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Is a self-adjoint operator continuous on its domain? Let $H$ be a Hilbert space, and $A : D(A) \subset H \rightarrow H$ be an unbounded linear operator, with a domain $D(A)$ being dense in H.
We assume that $A$ is self-adjoint, that is $A^*=A$.
Since $A$ is unbounded, we can find a sequence $x_n$ in the domain such tha... | If $A$ is unbounded, that means that for any $n$ there exists a $v\in D(A)$ such that $\|Av\|> n\|v\|$. Dividing such a $v$ by $\|v\|$, we may assume $\|v\|=1$, so we can find a sequence $(v_n)$ of unit vectors in $D(A)$ such that $\|Av_n\|>n$ for each $n$. The sequence $(v_n/n)$ then converges to $0$ but $Av_n/n$ do... | {
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Average $y$ from a range of $x$ in a parabola Given a parabolic/quadratic formula such as $ax^2 + bx + c =y$, how do I get the average value of $y$ given a range of $x$ ($x_{min}$ to $x_{max}$).
Real world example: if my formula represents the trajectory arc of a thrown object under the force of gravity, where $x$ is t... | The average of a function $f(x)$ in the interval $a$ to $b$ is given as
$$\frac{1}{b-a} \int_a^b f(x) dx$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Is there any integral for the Golden Ratio? I was wondering about important/famous mathematical constants, like $e$, $\pi$, $\gamma$, and obviously the golden ratio $\phi$.
The first three ones are really well known, and there are lots of integrals and series whose results are simply those constants. For example:
$$ \p... |
$$\int_{0}^{\phi}(1-x+x^2)^{1/\phi}(1-\phi^2x+\phi^3x^2)\mathrm dx=2^{\phi}\cdot\phi$$
A bit over-crowed in term of $\phi$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "166",
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"answer_id": 25
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Infinite intersection of open sets need not be open The following is the property of an open set:
The intersection of a finite number of open sets is open.
Why is it a finite number? Why can't it be infinite?
| Let $U_n=(-1/n,1/n)$ for $n=1,2,3,\dots$. Then $U_n$ is open for all $n$. Suppose $x\in\cap_n U_n$. Then if $x>0$ there is an $n$ such that $\frac1n<x$ (since $\frac1n\to0$). Thus $x\not\in U_n$ for that $n$. Same if $x<0$. Thus $x\not\in\cap_n U_n$. Now $0\in U_n$ for all $n$. Thus $\cap_n U_n=\{0\}$ which is no... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1654031",
"timestamp": "2023-03-29T00:00:00",
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I don't understand how Kirchhoff's Theorem can be true Kirchhoff's Matrix-Tree theorem states that the number of spanning trees of a graph G is equal to any cofactor of its Laplacian matrix.
Wouldn't this imply that all cofactors of a Laplacian matrix must be the same, as otherwise we could get a different number of sp... | I think you've made a mistake computing the Laplacian matrix.
I find \begin{bmatrix}2 & -1 &0 &-1 \\ -1 & 2 & -1 & 0 \\ 0 & -1 & 2 & -1 \\ -1 & 0 & -1 & 2 \end{bmatrix}
You can check that any cofactor has determinant $4$, which you can check by inspection is the right number of trees.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Find $\lim_{n\to\infty} \frac{(n!)^{1/n}}{n}$.
Find $$\lim_{n\to\infty} \frac{(n!)^{1/n}}{n}.$$
I don't know how to start. Hints are also appreciated.
| Let $f_n=\frac{n!}{n^n}$ so that $f_{n+1}= \frac{(n+1)!}{(n+!)^n+1}$
So $\frac{f_{n+1}}{f_n}=(1+\frac{1}{n})^{-n}$
Therefore
$\lim_{n\to\infty}(1+\frac{1}{n})^{-n}=\frac{1}{e}>0 $
We know that if $<f_n>$ is a sequence such that $f_n$ is greater than 0 for all n, and $\lim_{n\to\infty}\frac{f_{n+1}}{f_n}=l, l>0$. Then ... | {
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A milkman has $80\%$ of milk in his stock of $800$ litres of adulterated milk. How much $100\%$ milk to be added to give certain purity? Problem: A milkman has $80\%$ of milk in his stock of $800$ litres of adulterated milk. How much $100\%$ milk to be added to it so that the purity of milk is between $90\%$ and $95\%$... | $80\%$ is milk out of $800$ litres. That gives - you have $640$ litres of pure milk. Now, $640+x\over {800+x}$$>0.9$ That gives $x>800$ litres. Since, it should also be less than $0.95$ , You get $x<1200$ litres.
| {
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If $(x_1-a)(x_2-a)\cdots(x_n-a)=k^n$ prove by using the laws of inequality that $x_1x_2 \cdots x_n\geq (a+k)^n$
If $x_i>a>0$ for $i=1,2\cdots n$ and $(x_1-a)(x_2-a)\cdots(x_n-a)=k^n$, $k>0$, prove by using the laws of inequality that $$x_1x_2 \cdots x_n\geq (a+k)^n$$.
Attempt:
If we expand $(x_1-a)(x_2-a)\cdots(x_n-a... | Using convenient notation, we will prove a theorem from which the result of the question may be easily derived. Given any positive real numbers $ x_1, x_2,...,$ we will write their initial geometric means as $$g_n:=(x_1\cdots x_n)^{1/n}\quad(n=1,2,...).$$
Theorem.$\quad$Given any $a\geqslant0$, and for each $n$,
the ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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"answer_id": 0
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Infinite dimensional topological vectorspaces with dense finite dimensional subspaces Consider $\mathbb R$ as a $\mathbb Q$ vector space. Using the usual metric on $\mathbb R$, we find:
*
*$\mathbb Q \subset \mathbb R$ is dense and one dimensional (indeed every non-zero subspace appears to be dense).
*$\mathbb R$ i... | (1) Note that the topology on a finite-dimensional Hausdorff $\mathbf R$-vector space is uniquely determined and is always completely metriziable. As a complete metrizable space, is it closed in every topological vector space is it embedded into. Hence, finite-dimensional subspaces of Hausdorff topological vector space... | {
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Analyticity of $\tan(z)$ and radius of convergence Define $\tan(z)=\dfrac{\sin(z)}{\cos(z)}$
Where is this function defined and analytic?
My answer:
Our function is analytic wherever it has a convergent power series. Since (I am assuming) $\sin(z)$ and $\cos(z)$ are analytic the quotient is analytic wherever $\cos(z) \... | The Laurent series of $\tan z$ for $z\in \mathbb{C}$ is complicated:
$\displaystyle \tan z=z+\frac{z^{3}}{3}+\frac{2}{15}z^{5}+\ldots +
(-1)^{n-1}\frac{2^{2n}(2^{2n}-1)B_{2n}}{(2n)!}z^{2n-1}+\ldots \:, \: |z|<\frac{\pi}{2}$
where $B_{n}$ is Bernoulli number. The radius of convergence can be justified by $$4\sqrt{n\pi}... | {
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How lim sups here converted to appropriate limits? This question is from Rudin's Principles of Mathematical Analysis :
Consider the sequence $\{a_n\}$: $\{\frac 12, \frac 13, \frac 1{2^2}, \frac1{3^2}, \frac 1{2^3}, \frac1{3^3},\frac 1{2^4}, \frac1{3^4},\dots\}$
so, $$\limsup_{n\rightarrow \infty}\frac{a_{n+1}}{a_n... | Assuming the sequences start off with $n = 0$, we have $$\frac {a_{2k+1}}{a_{2k}} = \frac{\frac 1 {3^k}}{\frac 1 {2^k}} = \left(\frac 2 3\right)^k \tag{1}$$
$$\frac {a_{2k+2}}{a_{2k+1}} = \frac{\frac 1 {2^{k+1}}}{\frac 1 {3^k}} = \frac 1 2\left(\frac 3 2\right)^k\tag{2}$$
These two sequences together contain every elem... | {
"language": "en",
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Distance in metric space, triangle inequality problem Let $(X, d)$ be a metric space. Let $t\in (0,1]$. Show that $d^t: X\times X\to\mathbb{R}$
$$d^t (x,y) := d(x,y)^t, \forall x,y\in X$$
is also a distance function.
Problematic bit is the triangle inequality, when $0<t<1$
$$d (x,y)^t\leq d (x,z)^t+d (z,y)^t$$
Not sure... | I think I got it now (hope I'm not confusing things again). Setting $a := d(x,z)/d(x,y), b:= d(z,y)/d(x,y)$ the problem reduces to showing
$$ a^t + b^t \geq 1$$
If $0 < x < 1$, for $0 < t < 1$ we have $ x < x^t$, which gives
$$ \left(\frac{a}{a+b}\right)^t + \left(\frac{b}{a+b}\right)^t \geq \left(\frac{a}{a+b}\right) ... | {
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How many integer-sided right triangles are there whose sides are combinations?
How many integer-sided right triangles exist whose sides are combinations of the form $\displaystyle \binom{x}{2},\displaystyle \binom{y}{2},\displaystyle \binom{z}{2}$?
Attempt:
This seems like a hard question, since I can't even think of... | Solving $(1)$ for $z$, we have,
$$z = \frac{1\pm\sqrt{1\pm4w}}{2}\tag3$$
where,
$$w^2 = (x^2-x)^2+(y^2-y)^2\tag4$$
It can be shown that $(4)$ has infinitely many integer solutions. (Update: Also proven by Sierpinski in 1961. See link given by MXYMXY, Pythagorean Triples and Triangular Numbers by Ballew and Weger, 1979.... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Determining certain units in a local ring. I've been stuck on this problem for a while:
Let R be a commutative ring with $1 \neq 0$. If R has a unique maximal ideal (i.e. R is local), then either $x$ or $1-x$ (or both) are units in R.
| Suppose $I=(x)$ and $J=(1-x)$ are proper ideals.
Every proper ideal is contained in a maximal ideal, but there is only one, say $M$. Then $I \subset M$ and $J \subset M$ and so $x, (1-x) \in M \implies x+(1-x)=1 \in M$ which is absurd.
Then $I$ or $J$ or both are not proper, e.g. $I=R \implies \exists a \in R$ such tha... | {
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"timestamp": "2023-03-29T00:00:00",
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Finding the expected time for a stock to go from $\$25$ to $\$18$ given there is a support level at $20 with upward and downward biases. This problem is adapted from Stochastic Calculus and
Financial Applicationsby J. Michael Steele,
Springer, New York, 2001, Chapter 1, Section 1.6, page 9.
Consider... | Let $\tau=\inf\{n>0: Y_{n+1}<Y_n\mid Y_0=25\}$. Then
$$\mathbb E[\tau] = 1 + \frac23 + \frac13\mathbb E[\tau+1]\implies \mathbb E[\tau] =3.$$
It follows that $T_{21,20}=\tau=3$ and $T_{25,20}=5\tau=15$. The quantities $T_{20,19}$ and $T_{19,18}$ may be computed by a similar argument, and $$T_{25,18} = T_{25,20}+T_{20,1... | {
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Can Anyone solve this number of cases problem? There are $n$ different chairs around the round table, $C_1, C_2, ....C_n$, and one person going to give a number to each chair, $1$, $-1$, $i$ and $-i$. But $1$ can't be placed next to $-1$, and $i$ can't placed next to $-i$. And how can I get the number of the possible c... | As an approximation, start with the number of ways to make a line of $n$. You have $4$ choices for $C_1$ and $3$ choices for all the rest, so $4\cdot 3^{n-1}$. When you try to bend these into a circle, you will fail $\frac 14$ of the time, so so for a circle it is $3^n$
This is clearly not quite right as the end num... | {
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For which values $a, b \in \mathbb{R}$ the function $u(x,y) = ax^2+2xy+by^2$ is it the real part of a holomorphic function in $\mathbb{C}$
For which values $a, b \in \mathbb{R}$ the function $$u(x,y) =
ax^2+2xy+by^2$$ is the real part of a holomorphic function in
$\mathbb{C}$.
I think we have to take Cauchy-Riema... | This Result and Cauchy Riemann Equations shows that $u(x,y)$ is real part of holomorphic function iff $u$ is harmonic.
So,$u_{xx}+u_{yy}=0$ i.e. $b=-a$. QED
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Eigenfunctions of the Dirichlet Laplacian in balls I am trying to find out about the Dirichlet eigenvalues and eigenfunctions of the Laplacian on $B(0, 1) \subset \mathbb{R}^n$.
As pointed out in this MSE post, one needs to use polar coordinates, whence the basis eigenfunctions are given as a product of solutions of B... | Radial eigenfunctions are not zero at the centre of the ball, while nonradial eigenfunctions are zero there.
Indeed, recall that the radial part of any basis eigenfunction is
$$
r^\frac{2-n}{2} J_{l-\frac{2-n}{2}}(\sqrt{\lambda} r),
$$
where $\lambda$ is a corresponding eignevalue.
The parameter $l \in \{0,1,\dots\}$ c... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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problem solving PDE: $u*u_x + u_y = -u$ $$u*u_x + u_y = -u$$
$$u(0,t) = e^{-2t}$$
I tried solve with Lagrange and got 2 surfaces
$\phi(x,y,u) = x+u$ and $\psi(x,y,u) = y+ln(u)$ . when I used $u(0,t) = e^{-2t}$, I got a solution $$ u(x,y) = \dfrac{(1+\sqrt{1+4xe^{2y}})*e^{-2y}}{2} $$
but that seem to be wrong solution. ... | Solving it with the method of charateristics, I obtained the general solution on implicit form :
$$\Phi\left(x+u\:,\:y+\ln(u)\right)=0$$
which is consistent with your result: $\phi(x,y,u) = x+u$ and $\psi(x,y,u) = y+ln(u)$
$\Phi(\phi,\psi)$ is any derivable function of two variables. An equivalent form is :
$$x+u=F\lef... | {
"language": "en",
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Triangle angles.
For $\vartriangle$ABC it is given that $$\frac1{b+c}+\frac1{a+c}=\frac3{a+b+c}$$ Find the measure of angle $C$.
This is a "challenge problem" in my precalculus book that I was assigned. How do I find an angle from side lengths like this? I have tried everything I can. I think I may need to employ the... | Short answer:
According to the problem, the solution is unique, so any triple of values that satisfies the equation provides a solution. We immediately see that $a=b=c=1$ is a solution, hence the angle is 60 degrees.
Medium answer.
What count are the ratios between sides. So we can assume that $c=1$.
The equation is s... | {
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Steps for solving a simple quotient integral containing a product in the denominator. lets say that I have these two integrals:
$\int \frac{1}{e^{2x}-2e^x-3} \, dx$
and
$\int \frac{1}{(x+1)(x+2)(x-3)} \, dx$
I do recognize some properties and antiderivatives involved but wasn't successful by applying $u$-Substitution... | For the first integral write it as follows:$$\int \frac { dx }{ e^{ 2x }-2e^{ x }-3 } \, =\int { \frac { dx }{ \left( { e }^{ x }-3 \right) \left( { e }^{ x }+1 \right) } } =\frac { 1 }{ 2 } \left[ \int { \frac { dx }{ { e }^{ x }-3 } -\int { \frac { dx }{ { e }^{ x }+1 } } } \right] $$
then substitute:${ e }^{ x... | {
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Why is $\frac{(e^x+e^{-x})}{2}$ less than $e^\frac{x^2}{2}$? I have read somewhere that this equality holds for all $x \in \mathbb {R}$. Is it true, and if so, why is that?
$$\frac{(e^x+e^{-x})}{2} \leq e^\frac{x^2}{2}$$
| The taylor series on the LHS is $$\sum_{n=0}^{\infty} \frac{x^{2n}}{(2n)!}$$
The taylor series on the RHS is $$\sum_{n=0}^{\infty} \frac{(x^2/2)^n}{n!} = \sum_{n=0}^{\infty} \frac{x^{2n}}{(2n)!!}$$
Where $(2n)!!$ is the double factorial $(2 \times 4 \times \ldots \times 2n)$. It is easy to then see that $(2n)!! \leq (2... | {
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} |
Planarity on $10$ vertices Is there a planar graph on $10$ vertices such that its complement is planar as well?
I have troubles deciding if this is an elementary or deep question. By some other thread, the answer is an easy No for $11$ or more vertices.
| I think the answer is no, but I have only an experimental evidence.
$K_{10}$ has 45 edges, and the max number of edges for a planar graph with 10 vertices is $3\cdot10-6=24$.
So the only possible pairs of number of edges are {24, 21} and {23, 22}. I used Brendan McKay's program plantri to generate all the planar grap... | {
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"timestamp": "2023-03-29T00:00:00",
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Prove that if $ 2^n $ divides $ 3^m-1 $ then $ 2^{n-2} $ divides $ m $ I got a difficult problem. It's kind of difficult to prove. Can you do it?
Let $ m,n\geq 3 $ be two positive integers. Prove that if $ 2^n $ divides $ 3^m -1$ then $ 2^{n-2} $ divides $ m $
Thanks :-)
| Because $n \geq 3$ we get $8 \mid 3^m-1$ and so $m$ must be even .
Let $m=2^l \cdot k$ with $k$ odd . Now use the difference of squares repeatedly to get :
$$3^m-1=(3^k-1)(3^k+1)(3^{2k}+1)\cdot \ldots \cdot (3^{2^{k-1} \cdot l}+1)$$
Each term of the form $3^s+1$ with $s$ even has the power of $2$ in their prime factori... | {
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"source": "stackexchange",
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Question related to Integration and Probability Density Functions My question is regarding integration questions related to the probabilities of continuous random variables.
If X = 0 to 5 is represented by f1(x) and X=5 to 10 is represented by f2(x) and we want P(0<=X<=10). Would the answer be integral over 0 to 5 for... | Yes to your first question: You have it right about integrating each function in the range in which it applies. For example:
$$
f(x) =\left\{ \begin{array}{cc} \frac{x^2}{144} & 0\leq x<6 \\
\frac{x}{64} & 6 \leq x \leq 10 \end{array} \right.
$$
Then
$$P(2<x<7) = \int_2^6 \frac{x^2}{144} dx + \int_6^7 \frac{x}{64}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1657231",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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PDE, change of variables and differential operator "transformation" Given the wave equation:
$$ \frac{\partial^2 f}{\partial t^2} = c^2 \frac{\partial^2 f}{\partial x^2} $$
I change variables in this way:
$$ \xi = x+ct \\ \eta=x-ct $$
And the differential operators transform:
$$ \frac{\partial }{\partial x} = \frac{\... | Define $F(\xi,\eta)=f(t,x)$. Then, with some abuse of notation,
$$
\frac{\partial}{\partial x}f(t,x)=\frac{\partial}{\partial x}F(\xi,\eta)=\frac{\partial F}{\partial \xi}\frac{\partial\xi}{\partial x}+\frac{\partial F}{\partial \eta}\frac{\partial\eta}{\partial x}=\frac{\partial F}{\partial \xi}+\frac{\partial F}{\par... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Show $x^2 + 2\cos(x) \geq 2$ How do I show that $x^2 + 2\cos(x) - 2$ is always nonnegative (x is measured in radians)? If $x \geq 2$ or $x \leq -2$ then obviously, $x^2 \geq 4$, and so it must be true. But otherwise, $2\cos(x)$ can be as small as $-2$ and it is quite surprising that something that could potentially be ... | Since the functions are even, we only consider the region [0,2].
For $x= 0$, we have $2\ge 2$ is true.
For $x > 0$, we show the derivative of the function is positive. That is the function is increasing.
$2x - 2 \sin(x) = 2(x-\sin(x)) \ge 0$ as $x-\sin(x) \ge 0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1657439",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Prove $f$ is periodic if $ \int_{a}^b f(x)dx = 2 $ and $ \int_{y}^z f(x)dx = 1 $ $z,y\in (a,b)$ and $z-y=(a-b)/2$ if $ \int_{a}^b f(x)dx = 2 $ and for every $y,z \in (a,b)$ ($y$ smaller) such that $z-y = (a-b)/2$
we have $ \int_{y}^z f(x)dx = 1 $ how do we prove f is periodical and find the period?
| Let's assume $a<b$. By "periodic(al)" I assume you mean that:
$$f(x) = f(x+(b-a)/2) \quad \forall x \in [a, a + (b-a)/2]$$
Case 1:
If $f$ is not necessarily continuous, then the result is not true. That is because you can take any $f$ that works, then change its value on a finite number of points so that the new func... | {
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How to solve this indefinite integral using integral substitution? So while working on some physics problem for differential equations, I landed at this weird integral $$ \int \frac 1 {\sqrt{1-\left(\frac 2x\right)}}\,dx $$
So since there is a square root, I thought I could use trig substitution, but I couldn't find a... | HINT:
Make the substitution $x=2\sec^2(\theta)$ and arrive at
$$4\int \sec^3(\theta)\,d\theta$$
| {
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Solutions to the wave equation can be represented by a sine function? Consider the one dimensional wave equation: $$\frac{\partial^2 f(x, t)}{\partial t^2} - c^{2}\frac{\partial^2 f(x, t)}{\partial x^2} = 0. $$
I understand that one may find "wavy" solutions to this equation. But, $f(x, t) = x$ is a solution and it's ... | Let $u_1(x, t) = f_0(x -\sqrt{b} t)$ and $u_2(x, t) = f_0(x + \sqrt{b} t)$, we can verify that $u_1(x,t)$ and $u_2(x,t)$ both satisfy the wave equation. The general solution is $u(x, t) = a u_1(x, t) + b u_2(x,t)$. The solution represents the wave front (at the beach, facing ocean, and let the time stop, the wave in fr... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How many possible combinations in $7$ character password? The password must be $7$ characters long and it can include the combination: $10$ digits $(0-9)$ and uppercase letters $(26)$.
My Solution:
Thus in total there are $7$ slots, each slot could be either $0-9$ or $26$ letters $= 36$ possibilities for each slot, the... | You are correct. $36^7$ is the right answer.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Does the axiom of choice have any use for finite sets? It is well known that certain properties of infinite sets can only be shown using (some form of) the axiom of choice. I'm reading some introductory lectures about ZFC and I was wondering if there are any properties of finite sets that only hold under AC.
| There are two remarks that may be relevant here.
(1) This depends on what you mean by "finite sets". Even for (infinite setts of) pairs the axiom of choice is does not follow from ZF if one looks at an infinite collection. This is popularly known as the "pairs of socks" version of AC which is one of the weakest ones.
(... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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limit of a sequence when something about the limit is given Let $a_n$ be a sequence of real numbers such that
$$\lim_{n\to\infty}|a_n+3((n-2)/n)^n|^{1/n}=\frac35. $$
Then what is $\lim_{n\to\infty}a_n$?
| Call $b_n=((n-2)/n)^n$. You should know that $b_n \to e^{-2}$.
Now, you have for $n$ big enough
$$|a_n + b_n|^{1/n} < \frac 45$$
or equivalently,
$$|a_n + b_n| < \left( \frac 45 \right)^n \to 0$$
so by the squeeze theorem
$$\lim_n (a_n + b_n) = 0$$
Hence,
$$\lim_n a_n = - \lim_n b_n = -e^{-2}$$
| {
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Solve $3 = -x^2+4x$ by factoring I have $3 = -x^2 + 4x$ and I need to solve it by factoring. According to wolframalpha the solution is $x_1 = 1, x_2 = 3$.
\begin{align*}
3 & = -x^2 + 4x\\
x^2-4x+3 & = 0
\end{align*}
According to wolframalpha $(x-3) (x-1) = 0$ is the equation factored, which allows me to solve it, but... | \begin{align}x^2-4x+3&=x^2-3x-x+3\\
&=x(x-3)-(x-3)\\
&=(x-1)(x-3)
\end{align}
Note: You could also see that the sum of coefficients is zero, hence one root is $x=1$. Now divide the quadratic by $x-1$ to get the other factor.
| {
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"question_score": "1",
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How to find Fourier sine series of $f(x)=x(1-x), 0\lt x \lt 1$? How to find Fourier sine series of $f(x)=x(1-x), 0\lt x \lt 1$?
This is not an odd functions, so how to proceed?
| Define $g(x)=f(x)$ where $0<x<1$ and $g(x)=-f(-x)$ for $-1<x<0$. Then $g$ is an odd function. So you have to expand $g$ and that's the same as expanding $f$ in a sin series.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1658417",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Geometry proof, lost and need it explained please A rational number of the form $\frac{a}{2^{n}}$ (with $a,n$ integers) is called dyadic. In the interpretation, restrict to those points which have dyadic coordinates and to those lines which pass through several dyadic points. The incidence axioms, the first three betwe... | Outline: Consider the triangle with vertices $(0,0)$, $(1,0)$, and $(0,3)$. The line $3x+y=1$ meets one side of this triangle at $(0,1)$, but does not meet another side of this triangle.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finite length $A$-algebras are finitely generated? Let $k$ be a field and $M$ a module over a (associative, unital) finite-dimensional $k$-algebra $A$.
The length of $M$ is the unique length of a composition series for $M$.
How does $M$ having finite length imply that $M$ is finitely generated as an $A$-module?
I know ... | A module of finite length is noetherian, so it is finitely generated.
| {
"language": "en",
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If $H$ fixes three points, then could the normalizer of $H$ induce an orbit of size two on the fixed points Let $G$ be a transitive permutation group of degree $\ge 5$ acting such that every four-point stabilizer is trivial. Equivalently this means that every nontrivial element has at most three fixed points. Now if $1... | Yes this is possible. An example is ${\rm PSL}(3,2)$ in its natural action on $7$ points, with point stabilizer isomorphic to $S_4$, and $H$ a subgroup of order $2$ that does not lie in the derived subgroup of $G_\alpha$. Then $|N_G(H)|=8$, $|N_{G_\alpha}(H)|=4$.
| {
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If $x^2+y^2-xy-x-y+1=0$ ($x,y$ real) then calculate $x+y$ If $x^2+y^2-xy-x-y+1=0$ ($x,y$ real) then calculate $x+y$
Ideas for solution include factorizing the expression into a multiple of $x+y$ and expressing the left hand side as a sum of some perfect square expressions.
| I'll assume $x$ and $y$ are supposed to be real.
Let $s=x+y$ and $p=xy$; then your equation becomes
$$
s^2-2p-p-s+1=0
$$
or
$$
p=\frac{s^2-s+1}{3}
$$
The equation
$$
z^2-sz+p=0
$$
must have real roots; its discriminant is
$$
s^2-4p=-\frac{(s-2)^2}{3}\le0
$$
so we have $s=2$ (and $p=1$).
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Order preserving bijection from $\mathbb{Q}$ to $\mathbb{Q}\backslash\lbrace{0}\rbrace$
*
*How can one prove the existence of an order preserving bijection from $\mathbb{Q}$ to $\mathbb{Q}\backslash\lbrace{0}\rbrace$?
*Can you give an example of such a bijection?
| Choose an irrational number $\alpha$.
Let $x_1, x_2, \ldots$ be a strictly increasing sequence of rational numbers that converge towards $\alpha$.
Let $y_1, y_2, \ldots$ be a strictly decreasing sequence of rational numbers that converge towards $\alpha$.
Then define $f:\mathbb Q\to\mathbb Q\setminus\{0\}$ as:
*
*$f... | {
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"timestamp": "2023-03-29T00:00:00",
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solve $ax^3+by^3+cz^3+dx^2y+ex^2z+fxy^2+gxz^2+hy^2z+iyz^2=0$ for all triplets $(x,y,z)$. let $x,y,z$ be any 3 positive integers. If for all $x,y,z$, we have :
$$ax^3+by^3+cz^3+dx^2y+ex^2z+fxy^2+gxz^2+hy^2z+iyz^2=0$$ What can be said about the integral coefficients $a,b,c,e,f,g,h,i$?
I think they must be all zeros. Wha... | A special case of the parametrization:
$${z}^{3}-2\,a\,y\,{z}^{2}-b\,x\,{z}^{2}+\left( {a}^{2}-b\right) \,{y}^{2}\,z+\left( a\,b-3\,c\right) \,x\,y\,z+a\,c\,{x}^{2}\,z+\left( c+a\,b\right) \,{y}^{3}+\left( a\,c+{b}^{2}\right) \,x\,{y}^{2}+2\,b\,c\,{x}^{2}\,y+{c}^{2}\,{x}^{3}=c\cdot\,\left( {x}_{1}^{3}\,{c}^{2}+2\,{x}_{... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Category-Theoretic relation between Orbit-Stabilizer and Rank-Nullity Theorems In linear algebra, the Rank-Nullity theorem states that given a vector space $V$ and an $n\times n$ matrix $A$,
$$\text{rank}(A) + \text{null}(A) = n$$
or that
$$\text{dim(image}(A)) + \text{dim(ker}(A)) = \text{dim}(V).$$
In abstract algeb... | The intuition behind this question is spot-on. I'm going to try to fill out some of the details to make this work.
The first thing to note is that a linear map $A:V\to V$ also gives a genuine group action: it is the additive group of $V$ acting on the set $V$ by addition. That is, any $v\in V$ acts on $x\in V$ as $v: x... | {
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What is a single-prime function other than $f(x)=x!$? [Noob warning]: I am not a mathematician. If you use jargon, please explain or reference.
Other than $f(x)=x!$, what is a univariate non-piecewise function with a domain that is either all integers, or an infinite-sized subset of all integers, and whose range contai... | $$f(x)=x^2+x$$
More generally, if $g(x)$ is any function with $f(1)=p$ is prime and $f(x) \geq 2$ for all $x$ then
$$f(x)=xg(x)$$
Also
$$h(n)= lcm [1,2,3,..,n]$$
Also
$$u(n)=n^{n-1}$$
| {
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Is the group $G$ always isomorphic to the group $G/N \times N$?
Let $N$ be a normal subgroup of $G$. Is the group $G$ always
isomorphic to the group $G/N \times N$?
I don't think this is true but I can't think of a counter-example. What's an easy counter-example (or way to prove the contrary)?
| Consider the cyclic group of order $4$, say $C_{4}$. It has a nornal subgroup $H$ of index 2. $C_{4}/H$ is is a cyclic group of order $2$, isomorphic to $H$! But $H \times H$ is not cyclic!
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What is wrong with my integral? $\sin^5 x\cos^3 x$ I am trying to do an integration problem but am running into a problem! My answer is different from what the solution says. My attempt:
Evaluate $$\int\cos^3x\sin^5 x\mathop{dx}$$
$\int\cos^3x\sin^5 x\mathop{dx}=(1-\cos^2 x)^2\sin x \cos ^3 x$
Then with u substitutio... | Let's see a simpler case:
$$
\int 2\sin x\cos x\,dx
$$
You have two choices: either do $u=\sin x$, so $\cos x\,dx=du$, and you get
$$
\int 2u\,du=u^2+c=\sin^2x+c
$$
or do $v=\cos x$, so $\sin x\,dx=-dv$, and you get
$$
\int -2v\,dv=-v^2+c=-\cos^2x+c
$$
Which one is right? Both, of course, but this doesn't mean you reac... | {
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"timestamp": "2023-03-29T00:00:00",
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Effect of row augmentation on value of determinant.
Part(a) is done.
How to proceed for part (b).
My first question is what do they mean by row augmentation ?
Do they mean the row operation of adding k times the first row to third by row augmentation ?
| An idea: The row augmentation on $\;A\;$ is then the same as the product $\;GA\;$. Now, why not using the all important theorem that for any two square matrices $\;X,Y\;$ of the same order we have that $\;\det(XY)=\det X\cdot\det Y\;$ ?
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Explain $\tan^2(\tan^{-1}(x))$ becoming $x^2$ How does $\tan^2(\tan^{-1}(x))$ become $x^2$?
I feel that the answer should contain a tan somewhere and not just simply $x^2$. "Why?" you might ask, well I thought that $\tan^2(\theta)$ was a special function that has to be rewritten a specific way.
| Because $\tan^{-1}x $ is not the reciprocal of $\tan x$, but the inverse function $\arctan x$.
| {
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Simple way to estimate the root of $x^5-x^ 4+2x^3+x^2+x+1=0$ How to give a mathematical proof that for all complex roots of $x^5-x^ 4+2x^3+x^2+x+1=0$, their real part is smaller than 1, and there is at least one root whose real part is larger than 0. If possible, not to solve any algebraic equation whose degree is larg... | Using the Routh-Hurwitz stability criterion you can tell how may roots of your system are in the open left-hand complex plane - i.e., the set $\{z\in\mathbb{C}: \operatorname{Re}(z) < 0\}$.
In your case for the polynomial $p(x)=x^5 - x^4 + 2x^3 + x^2 + x + 1$ the Hurwitz matrix is:
1.0000 2.0000 1.0000
-1... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Chain rule for composition of $\mathbb C$ differentiable functions
What are the different methods to formulate a version for chain rule for composition of $\mathbb C-$ differentiable functions? Give a short proof.
| "Differentiable" will mean "complex differentiable" below. I'll assume the fact that if a function has a complex derivative at a point, the function is continuous at that point.
Thm: Suppose $f$ is differentiable at $a$ and $g$ is differentiable at $f(a).$ Then $g\circ f$ is differentiable at $a$ and $(g\circ f)'(a)= ... | {
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The intervals $(2,4)$ and $(-1,17)$ have the same cardinality I have to prove that $(2,4)$ and $(-1,17)$ have the same cardinality.
I have the definition of cardinality but my prof words things in the most confusing way possible. Help!
| The general way to show two sets $X,Y$ have the same cardinality is to show that there is a function $f:X\rightarrow Y$ that is both 1) injective and 2) surjective. That is 1) for all $a\neq b\in X$ we must have $f(a)\neq f(b)$ and 2) for all $y\in Y$ there must exist $x\in X$ such that $f(x)=y$.
| {
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"answer_id": 1
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Linear independence of certain vectors of $\mathbf{C}^2$ over $\mathbf{R}$ Suppose that $\{e_1,e_2\}$ where $e_1=(1,0)$ and $e_2=(0,1)$ is the standard basis of $\Bbb C^2$ as a vector space over $\Bbb C$. Show that $\{e_1,ie_1,e_2,ie_2\}$ is a basis of $\Bbb C^2$ as a vector space over $\Bbb R$ and conclude that $\dim_... | As another user has described linear independence (and hopefully made it clear that $dim_{\mathbb{R}} \mathbb{C}^2 = 4 = 2\dim_{\mathbb{C}}\mathbb{C}^2$ ), you need now to show that $\{e_1,e_2,ie_1,ie_2\}$ spans $\mathbb{C}^2$. Let $\xi \in \mathbb{C}^2$ be arbitrary. Then, by our assumptions, there exists $x_1,x_2 \in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1659956",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Convergence of $\sum_{n=1}^\infty n^2 \sin(\frac{π}{2^n}) $ This is my series: $$\sum_{n=1}^\infty n^2 \sin(\frac{π}{2^n}) $$
WolframAlpha says it converges, but I have no idea how to get the answer. I have learned comparison test, ratio test, root test and integral test. I don't really know which one of those to use. ... | The following inequality holds:
$$
\sin x \le x\qquad(x\ge 0)
$$
Then
$$
0\le \sin\left(\frac{\pi}{2^n}\right)\le \frac{\pi}{2^n}.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1660049",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
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Evaluate the limit $\lim_{x\to \infty}( \sqrt{4x^2+x}-2x)$
Evaluate :$$\lim_{x\to \infty} (\sqrt{4x^2+x}-2x)$$
$$\lim_{x\to \infty} (\sqrt{4x^2+x}-2x)=\lim_{x\to \infty} \left[(\sqrt{4x^2+x}-2x)\frac{\sqrt{4x^2+x}+2x}{\sqrt{4x^2+x}+2x}\right]=\lim_{x\to \infty}\frac{{4x^2+x}-4x^2}{\sqrt{4x^2+x}+2x}=\lim_{x\to \infty}... | Hint :
$\displaystyle\lim_{x\to \infty}\frac{x}{\sqrt{4x^2+x}+2x}=\lim_{x\to \infty}\frac{1}{\sqrt{4+\frac{1}{x}}+2}$ , dividing numerator and denominator by $x$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1660120",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
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A normal subgroup of a nilpotent group intersects the centre non-trivially How could I prove the above statement?
For $G$ nilpotent and $N \lhd G$, how do I show that $N \cap Z(G) \neq 1$?
| Take the upper central series of the group
$$1=Z_0\le Z_1\le\ldots\le Z_n=G$$
Since $\;1\neq N\lhd G\;$ there exists $\;1\le k< n\;$ such that $\;N\cap Z_k=1\;$ but $\;N\cap Z_{k+1}\neq 1\;$ , so take commutator groups:
$$[G, N\cap Z_{k+1}]\le[G,N]\cap[G,Z_{k+1}]\le N\cap Z_k=1$$
because normality: a subgroup $\;N\le G... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1660303",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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(H.W question) In Mathemaical Analysis of Rudin example 1.1 Pg 2 The author went on and proved that
1) There exists no rational $p$ such that $p^2=2$
2) He defined two sets $A$ and $B$ such that if $p\in A$ then $p^2 <2$ and if $p\in B$ then $p^2>2$ and then constructed $$q=p - \frac{p^2-2}{p+2}$$ and $$q^2-2=\frac{2... | Rudin doesn't give a #*@$ whether there is a rational square root of 2 or not. What he's trying to show is that you can divide all the rational numbers into two sets that exhaust the rationals; that one set can have every element larger than every element in the other yet it is possible to have no limits to either set;... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1660377",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Diffeomorphism between an ellipse and unit circle? I'm trying to learn about diffeomorphism and an example asks for a diffeomorphism between an ellipse and an unit circle.
How does one construct such?
| You need to use a specific ellipse, and presumably the person asking the question would not accept the unit circle as an example of an ellipse (in which case the identity mapping would deliver).
I would suggest using the standard equation of an ellipse with horizontal and vertical axes, centered on the origin. This is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1660442",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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smallest number for given sum of digits I am trying to find the smallest number if we are given the sum of its digits. Suppose that sum of digits is 9 then it should be 9 instead of 18,36,63 and similarly if sum of digits is 11 then desired answer is 29 not 92 or any other number bigger than 29.I tried to write sum o... | In our so-called positional numeration system, the digits get a weight that increases from right to left, following the powers of ten (units, tens, hundreds, thousands...).
So to minimize the number you will allocate the budget in priority to the positions with the smallest weight.
This is why the solution is by puttin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1660540",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How do find if a relation is a function algebraically Is there a way to see if a relation is a function without having to do a "vertical line test" (where you draw a vertical line on the graph and if there line touches two points then it's not a function).
To determine if a function is even or odd you simply go f(x) = ... | For a relation to be a function, it must be one-to-one or injective, meaning that it must map each input into a different output.
If you can't use the vertical line test, see if you can determine whether or not the function/relation has branches.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1660603",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why are matrix norms defined the way they are? Given $A$ a square matrix
Define:
$\|A\|_1$ as the max absolute column sum
$\|A\|_2$ as the sum of the squares of each element
$\|A\|_\infty$ as the max absolute row sum
Pray tell, why are matrix norms defined this way? Is this a property inherited or derived from the vec... | Matrices can be considered as linear operators. And for a linear operator $A:X\to Y$, where $X,Y$ are normed spaces with norms $\|.\|_X,\|.\|_Y$, the definition of the operator norm is
$$\|A\|=\sup\limits_{x\in X,x\neq 0}{\frac{\|Ax\|_Y}{\|x\|_X}}$$
If you use this definition, then the obtained matrix norm is called i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1660720",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Use $R_4$ to estimate the area under the curve $y= \frac{2}{1+x^2}$ between $x=0$ and $x=1$. Question : Use $R_4$ to estimate the area under the curve $y= \frac{2}{1+x^2}$ between $x=0$ and $x=1$.
Not sure how to proceed with this question.
| Let
$$f(x)=\frac2{(1+x^2)}$$ Between $0$ and $1$ we have the width of each section equal to $\frac14$ because we are using $4$ subsections.
For the first subsection we have width times height so $\frac14\times f(0.25)$
($0.25$ because we are using $R$-approximations).
We increment the input for the function by $0.25$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1660950",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove $\frac{a^3+b^3+c^3}{3}\frac{a^7+b^7+c^7}{7} = \left(\frac{a^5+b^5+c^5}{5}\right)^2$ if $a+b+c=0$ Found this lovely identity the other day, and thought it was fun enough to share as a problem:
If $a+b+c=0$ then show $$\frac{a^3+b^3+c^3}{3}\frac{a^7+b^7+c^7}{7} = \left(\frac{a^5+b^5+c^5}{5}\right)^2.$$
There are,... | Let $T_{m}$ be $a^m+b^m+c^m$.
Let $k=-ab-bc-ca$, and $l=abc$.
Note that this implies $a,b,c$ are solutions to $x^3=kx+l$.
Using Newton's Identity, note the fact that $T_{m+3}=kT_{m+1}+lT_{m}$(which can be proved by summing $x^3+kx+l$)
It is not to difficult to see that $T_{2}=2k$, $T_3=3l$, from $a+b+c=0$.
From her... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1661035",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "34",
"answer_count": 7,
"answer_id": 4
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is there a Relationship between duplicity of EigenValue and dimension of it's EigenSpace? giving characteristic polynomial of a matrix (Which has eigenvalues with it's duplicity) how can we understand the dimension of eigenspace of each eigenvalue without direct calculation?
in addition, is there a relationship between... | The power of the eigenvalue in the characteristic polynomial is called algebraic multiplicity and the dimension of its eigenspace geometric multiplicity. One can show that
$$1 \leq \text{ geometric mult.} \leq \text{algebraic mult.}$$
always holds. Note that equality does not hold in general, e.g. take
$$\pmatrix{0 & ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1661174",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Show that $K_1\cap K_2\cap \dots,K_N$ is compact Let $K_1,K_2,\dots,K_N$ be compact subsets of the mectric space $(X,d)$.
Show that $K_1\cap K_2\cap \dots,K_N$ is compact.
My approach:
I think I should use the definition of compact sets in my textbook:
Let $(X,d)$ be a metric space. A subset $K\subseteq X$ is compact ... | Actually, arbitrary intersection (not only finite) of compact subsets is also compact. This is pretty easy to see: in metric spaces, compact subsets are closed, so their intersection is also closed. On the other hand, the closed subset of a compact set is also compact: suppose $K$ is compact and $N \subset K$ is closed... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Is the range of a self-adjoint operator stable by its exponential? Let $H$ be an Hilbert space, and $A \in L(H)$ be a bounded linear self-adjoint operator on $A$. We assume that $R(A)$, the range of $A$, is not closed.
Is it true or not that $R(A)$ is stable by $e^{-A}$? Meaning: if $y \in R(A)$ then $e^{-A} y \in R(A)... | Ok so I think I have a (partial) answer, by means of some spectral analysis.
Assuming that $A$ is self-adjoint, and that $H$ is separable, we can find a family $(\sigma_n)$ of decreasing nonnegative eigenvalues and $(u_n)$ of orthonormal eigenvectors such that for all $x \in H$, $Ax= \sum\limits_{n=0}^\infty \sigma_n \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1661501",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Choosing value of ω for SOR I am learning about successive overrelaxation, and I'm wondering if there is an intuitive reason as to why ω must be between 0 and 2. I know that the method will not converge is ω is not on this interval, but I'm wondering if anyone can give an explanation of why this makes sense.
| I shall try to give you an intuitive idea of why $\omega \in (0,2)$ is essential.
There are many different ways of stating the SOR iteration, but for the purpose of answering your question I will use the following form
\begin{equation}
x^{(k+1)} = (1 - \omega) x^{(k)} + \omega D^{-1} \left[ b - Lx^{(k+1)} - Ux^{(k)} \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1661622",
"timestamp": "2023-03-29T00:00:00",
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Show that the limit as x approaches zero for $\frac{2^{1/x} - 2^{-1/x}}{2^{1/x} + 2^{-1/x}}$ does not exist This problem was in one of the first chapters of a calculus text, so how would you go about solving this without applying L'Hôpital's rule?
I attempted factoring out $2^{1/x}$, as well as using u substitution for... | We can simplify the problem as follows:
$$\mathrm f(x) := \frac{2^{1/x}-2^{-1/x}}{2^{1/x}+2^{-1/x}} \equiv \frac{2^{2/x}-1}{2^{2/x}+1}
\equiv \frac{4^{1/x}-1}{4^{1/x}+1} \equiv \frac{1-4^{-1/x}}{1+4^{-1/x}}$$
All we need to do is consider the limits of $4^{1/x}$ and $4^{-1/x}$ as $x$ tends to zero.
*
*If $x<0$ and $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1661845",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$\lim_{x\to 0} (2^{\tan x} - 2^{\sin x})/(x^2 \sin x)$ without l'Hopital's rule; how is my procedure wrong?
please explain why my procedure is wrong i am not able to find out??
I know the property limit of product is product of limits (provided limit exists and i think in this case limit exists for both the functions)... | One can rewrite:
$$\frac{2^{\tan x}-2^{\sin x}}{x^2\sin x}=\frac{2^{\tan x}-2^{\sin x}}{\tan x - \sin x}\frac{\tan x - \sin x}{x^2 \sin x}=\frac{2^{\tan x}-2^{\sin x}}{\tan x - \sin x}\frac{1-\cos x}{x^2 \cos x}=\frac{2^{\tan x}-2^{\sin x}}{\tan x - \sin x}\frac{\sin^2 x}{x^2}\frac{1}{(1+\cos x)\cos x}$$
$$=2^{\sin x}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1661945",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 5
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Does existence of independent variables ensure that there are infinitely many solutions? As I stated above. In a system of equations, does existence of independent variables ensure that there are infinitely many solutions?
THANK YOU!
| I assume you mean independent variable as to mean the same as my $t$ in the following example:
Say we a system of equations
\begin{align}
-x_1-2x_2-5x_3&=-3 \\
2x_1+3x_2+8x_3&=4 \\
2x_1+6x_2+14x_3&=10 \\
\end{align}
which we write on matrix form and then reduce:
\begin{align}
\pmatrix{
-1 & -2 & -5 & -3 \\
2 & 3 & 8 &... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1662063",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to evaluate this limit? I have a problem with this limit, I have no idea how to compute it. Can you explain the method and the steps used(without L'Hopital if is possible)? Thanks
$$\lim _{x\to 0+}\left(\frac{\left[\ln\left(\frac{5+x^2}{5+4x}\right)\right]^6\ln\left(\frac{5+x^2}{1+4x}\right)}{\sqrt{5x^{10}+x^{11}}-... | Let's try the elementary way. We have
\begin{align}
L &= \lim _{x \to 0^{+}}\left(\dfrac{\left[\log\left(\dfrac{5 + x^{2}}{5 + 4x}\right)\right]^{6} \log\left(\dfrac{5 + x^{2}}{1 + 4x}\right)}{\sqrt{5x^{10} + x^{11}} - \sqrt{5}x^5}\right)\notag\\
&= \lim _{x \to 0^{+}}\left(\dfrac{\left[\log\left(1 + \dfrac{x^{2} - 4x}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1662144",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How can one do dimensional analysis when units are not known? In the sciences, we can do dimensional analysis and unit checks to verify whether or not the LHS and the RHS have the same units. If we have the following function:$$y=f(x)=x^{2}$$
what ensures the preserving of units? I have a feeling it is the exponent o... | If you're doing mathematics, usually you work with dimensionless quantities, so it makes no sense to try to do dimensional analysis.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1662200",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finding the limit of $\sqrt{9x^2+x} -3x$ at infinity To find the limit
$$\lim_{x\to\infty} (\sqrt{9x^2+x} -3x)$$
Basically I simplified this down to
$$\lim_{x\to\infty} \frac{1}{\sqrt{9+1/x}+3x}$$
And I am unaware of what to do next. I tried to just sub in infinity and I get an answer of $0$ , since $1 / \infty = 0$. H... | For $x>0$ we have
$$\sqrt{9x^2+x}-3x=\frac{\big(\sqrt{9x^2+x}-3x\big)\big(\sqrt{9x^2+x}+3x\big)}{\sqrt{9x^2+x}+3x}= \\ =\frac{9x^2+x-9x^2}{\sqrt{9x^2+x}+3x}=\frac{x}{\sqrt{9x^2+x}+3x}=\frac{1}{\sqrt{9+\frac{1}{x}}+3}=$$
Thus:
$$\lim_{x\rightarrow\infty}\big(\sqrt{9x^2+x}-3x\big)=\lim_{x\rightarrow\infty}\frac{1}{\sqrt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1662291",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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prove that $\sqrt{2}$ is not periodic. If I am asked to show that $\sqrt{2}$ does not have a periodic decimal expansion.
Can I just prove that $\sqrt{2}$ is irrational , and since irrational numbers are don't have periodic decimal expansions then I am done?
Thank you.
| Suppose $\sqrt{2}$ has a periodic decimal expansion, i.e. it has the form
$$\sqrt{2} = 1.a_1a_2 \cdots a_m \overline{d_1 d_2 \cdots d_n}$$
where the overline indicates the repeating digits. Then
$$10^m \sqrt{2} = 1a_1a_2 \cdots a_m . \overline{d_1 d_2 \cdots d_n}$$
and
$$10^{m+n} \sqrt{2} = 1a_1 a_2 \cdots a_m d_1 d_2 ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Determining the bounds for a triple integral? I need help with a triple integration problem. I don't need help actually integrating this thing, I just need help with setting the actual integral(s) up. Specifically, I dont know how to determine what the bounds are.
We basically have a 3D donut. The problem says we can m... | You have a problem of order: The limits on the $z$ integral depend on $r$, and therefore it has to be done where $r$ is defined. In other words, the integral is $$\int_0^{2\pi}\int_2^4\int_{-\sqrt{a^2-(r-c)^2}}^{\sqrt{a^2-(r-c)^2}} rdzdrd\theta$$
And you are correct that $c = 3$ and $a = 1$
But as noted in the comments... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1662498",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Elementary symmetric polynomial related to matrices I've encountered with the following question while reading something about invariant polynomial in Chern-Weil theory:
For a matrix $X \in M(n;\Bbb{R})$ ,denote its eigenvalues by $\lambda_1,...,\lambda_n$,and the $n$ symmetric polynomials by $\sigma_1,...\sigma_n$,tha... | You can actually just read this from the characteristic polynomial $$\chi(t)=\det(tI-X)=\prod_{k=1}^n(t-\lambda_k)=\sum_{k=0}^n(-1)^k\sigma_k(X)t^{n-k}\text{,}$$ because then your polynomial is $$\det(I+tX)=(-t)^n\chi(-t^{-1})=\sum_{k=0}^n\sigma_k(X)t^k\text{.}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1662615",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$X,Y$ infinite dimensional NLS , not both Banach , then $\exists T \in \mathcal L(X,Y)$ such that $R(T)$ is not closed in $Y$? Let $X,Y$ be infinite dimensional normed-linear spaces , not both Banach , then does there necessarily exist a continuous linear transformation $T:X \to Y $ such that $range (T)$ is not closed ... | Let $X$ be a Banach space and $Y$ the increasing union of a sequence of finite-dimensional spaces $F_n$. Then $X = \bigcup T^{-1}(F_n)$ with each $T^{-1}(F_n)$ closed. By the Baire category theorem, some $T^{-1}(F_n)$ has
nonempty interior, and therefore it is all of $X$. That is, $\text{Range}(T)$ is a subspace of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1662752",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Prove $\sinh(x)\leq 3|x|$ for $|x| < 1/2$ I need to prove $|\sinh(x)|\leq 3|x|$ for $|x| < 1/2$
My current progress states that
$$x \leq \frac{ (1+x)-(1-x) }{2} \leq \sinh(x)\leq \frac{ \frac{1}{1-x}-\frac{1}{1+x} }{2}$$
whereas
$$|\sinh(x)|\leq \max\left(|x|,\left|\frac{x}{1+x^2}\right|\right)\leq|x|+\left|\frac{x}{1+... | It seems you can use the series definition. Then
$$
\left|\sinh(x)\right|=|x|\left(1+\frac{|x|^2}{3!}+\frac{|x|^4}{5!}+…\right)
\le|x|·\left(1+\frac{|x|^2}{6}+\frac{|x|^4}{6^2}+…\right)
\\=|x|·\frac{6}{6-x^2}
$$
for $|x|<\sqrt{6}$ and for $|x|\le\frac12$ this can be reduced to
$$
|\sinh(x)|\le|x|·\frac{24}{23}.
$$
Actu... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A probability theory question about independent coin tosses by two players
Say Bob tosses his $n+1$ fair coins and Alice tosses her $n$ fair coins. Lets assume independent coin tosses. Now after all the $2n+1$ coin tosses one wants to know the probability that Bob has gotten more heads than Alice.
The way I thought ... | Get out some red paint. Paint all the heads sides on Bob's coins, and paint all the tails sides of Alice's coins. Bob wins if and only if at least $n + 1$ coins out of $2n + 1$ land red side up. By symmetry, the probability of this happening is $1/2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1662958",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 4,
"answer_id": 0
} |
How do I prove this seemingly obvious property of subgroups The statement is the following: Given an abelian group $G=\langle a_1,...,a_t\rangle$, and a subgroup $H$ of $G$, we need at most $t$ elements to generate $H$; i.e. $H=\langle b_1,...,b_t\rangle$ for some $b_1,...,b_t\in H$.
While this statement seems obvious ... | I already had a proof of this written down, so I have copied and pasted it.
Let $K \le G$ with $G$ an (additive) abelian group generated by $x_1,\ldots,x_n$. We shall prove by induction on $n$ that $K$ can be generated
by at most $n$ elements. If $n=1$ then $G$ is cyclic and hence so
is $K$. Suppose $n>1$, and let $H... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1663045",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Integrate $I= \int \frac{x^2 -1}{x\sqrt{1+ x^4}}\,\mathrm d x$
$$I= \int \frac{x^2 -1}{x\sqrt{1+ x^4}}\,\mathrm d x$$
My Endeavour :
\begin{align}I&= \int \frac{x^2 -1}{x\sqrt{1+ x^4}}\,\mathrm d x\\ &= \int \frac{x}{\sqrt{1+ x^4}}\,\mathrm d x - \int \frac{1}{x\sqrt{1+ x^4}}\,\mathrm d x\end{align}
\begin{align}\te... | From here and here we learn that (mistake 1)
\begin{align}
\int \frac{x}{\sqrt{1+ x^4}}dx&=\frac12\arcsin x^2\\
&=\frac12\ln(x^2+\sqrt{1+x^4})
\end{align}
Hence (for part 2 I basically take your solution multiplied by $-1$: mistake 2)
\begin{align}
I&=\frac12\ln x^2+\sqrt{1+x^4})-\frac14 \ln \frac{ \sqrt{1+ x^4}-1}{\sq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1663130",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Proof problem: show that $n^a < a^n$ for all sufficiently large n I would like to show that $n^a < a^n$ for all sufficiently large $n$, where $a$ is a finite constant.
This is clearly true by intuition/graphing, but I am looking for a rigorous proof. Can anyone help me out? Thanks.
| Take $\log$ from both sides $a \log n < n \log a $
Now lets check who grows faster $\lim\limits_{n\to \infty}\frac{n \log a}{a \log n}=\infty$ and $\lim\limits_{n\to \infty}\frac{a \log n}{n \log a}=0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1663333",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 2
} |
Initial form of a polynomial I am reading some tropical geometry and came up with the concept of the initial form of a polynomial. The definition says that the initial form of f with respecto to a weight vector $w \in \mathbb{R}^{n+1}$ is \begin{equation}
in_w(f) = \sum_{\substack{u\in \mathbb{N}^{n+1} \\
val(c_u) +... | Have a look at Remark 5.7 of Gublers "Guide to tropicalization", I find his approach easier to understand than the definition you seem to have copied from Sturmfels' book.
Initial forms can be used to define the tropicalization of a variety, and are especially important if the field K ist trivially valued.
The set of a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1663455",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Cauchy Residue Theorem Integral I have been given the integral $$\int_0^ {2\pi} \frac{sin^2\theta} {2 - cos\theta} d\theta $$ I have use the substitutions $z=e^{i\theta}$ |$d\theta = \frac{1}{iz}dz$ and a lot of algebra to transform the integral into this $$\frac{-i}{2} \oint \frac{1}{z^2}\frac{(z-1)^2}{z^2-4z+1}dz$$ I... | There was an error in the original post. We have
$$\int_0^{2\pi}\frac{\sin^2(\theta)}{2-\cos(\theta)}d\theta=-\frac i2\oint_{|z|=1}\frac{(z^2-1)^2}{z^2(z^2-4z+1)}\,dz$$
There are two poles inside $|z|=1$. The first is a second order pole at $z=0$ and the second is a first order pole at $z=r_2$.
To find the reside of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1663506",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
length of $\sqrt{x} + \sqrt{y} = 1$ ? on $[0,1]$ how to calcuate length of $\sqrt{x} + \sqrt{y} = 1$ ?
I put $x = \cos^4{t} , \ y = \sin^4{t}$ but failed.
Do you have Any good idea form here?
| $$
\sqrt{x}+\sqrt{y}=1\implies y'=-\frac{\sqrt{y}}{\sqrt{x}}
$$
Substituting $x\mapsto u^2$ and $2u-1\mapsto\tan(\phi)$ gives
$$
\begin{align}
\int_0^1\sqrt{1+y'^2}\,\mathrm{d}x
&=\int_0^1\sqrt{1+\frac yx}\,\mathrm{d}x\\
&=\int_0^1\sqrt{1+\frac{1+x-2\sqrt{x}}x}\,\mathrm{d}x\\
&=2\int_0^1\sqrt{2u^2-2u+1}\,\mathrm{d}u\\
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1663573",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Convert PI to base 4. Does my unique human genome exist in the sequence of digits? The human genome consists of sequences of BASE Pairs A G C T
Convert the number PI to base 4. Does my unique human genome exist in the sequence of digits?
| A heuristic argument would go as follows: Assume your genome $G$ is a string of $n\gg1$ digits over $\{0,1,2,3\}$. Denote by $x_k$ the $k^{\rm th}$ digit of $\pi$ in base $4$. For each $r\geq0$ the probability that
$$(x_{rn+1},x_{rn+2},\ldots,x_{rn+n-1}, x_{(r+1)n})\ne G$$
amounts to $(4^n-1)/ 4^n<1$. Therefore the pr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1663674",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Limit of probability density function as random variable approaches +/- infinity Consider a complex-valued function $\Psi(x,t)$ such that $|\Psi|^2$ is a probability density function for $x$ (for any time $t$).
In his introductory Quantum Mechanics book, David J Griffiths writes that the limit of the expression
$$\Psi^... | Define
$$f(x)=\begin{cases} 1-2x & x \in [0,1/2] \\ 2x+1 & x \in [-1/2,0] \\ 0 & \text{otherwise}\end{cases}.$$
Geometrically, this is a triangle with height $1$ and width $1$ centered at zero.
Now define
$$g(x)=\sum_{n=1}^\infty f(n^2(x-n))$$
This is a sequence of separate triangles with height $1$ and width $1/n^2$.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1663758",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What are the eigenvectors of 4 times the identity matrix? What are the eigenvectors of the identity matrix
$
I=
\left[ {\begin{array}{cc}
1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} } \right]
$
| Follow the definition. In order to determine the eigenvectors of a matrix, you must first determine the eigenvalues.
Substitute one eigenvalue λ into the equation Ax = λx or, equivalently, into (A − λI)x = 0 and solve for x; the resulting nonzero solutions form the set of eigenvectors of A corresponding to the selecte... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1663915",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
how to choose which one holds? let $X$ be any set with the property that for any two metrics $d_1$ and $d_2$ on $X$, the identity map
$id:($X$,d_1$)$\to ($X$,d_2)$ is Continuous.
which of the following are true?
1) $X$ must be a singleton.
2) $X$ can be any finite set.
3) $X$ cannot be finite.
4) $X$ may be infinite b... | It's easy if you know that all the norms over $\mathbb{R}$ are equivalent. Then you function is continuous if you choose $X=\mathbb{R}$ and $d_1$ $d_2$ metrics induced by norms over $\mathbb{R}$. In fact $ \exists c \in \mathbb{R}: d_2(x,y) \le c \ d_1(x,y)$ and if you take a succession $(x_n)$ of real numbers converge... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1664013",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Make $2^8 + 2^{11} + 2^n$ a perfect square Can someone help me with this exercise? I tried to do it, but it was very hard to solve it.
Find the value of $n$ to make $2^8 + 2^{11} + 2^n$ a perfect square.
It is the same thing like $4=2^2$.
| Here is a very late answer since I just saw the problem:
By brute force, we may check that $n=12$ is the smallest possible integer such that $2^8+2^{11}+2^n$ is a perfect square. We also claim that this is the only integer.
To see why the above is true, let $2^8+2^{11}+2^n=2^8(1+2^3+2^k)=2^8(9+2^k), k \ge 4$. Now, we o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1664088",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 8,
"answer_id": 4
} |
Determine every vector field such that its field lines are contour lines to $g(x, y) = x^2 + 4y^2$ Is it possible to determine every vector field such that it's field lines are contour lines to $g(x, y) = x^2 + 4y^2$? If so, how?
| At each point ${\bf z}=(x,y)$ the gradient $\nabla g(x,y)=(2x,8y)$ is orthogonal to the contour line of $g$ through ${\bf z}$. Turn this gradient by ${\pi\over2}$ counterclockwise to obtain the vector field
$${\bf v_*}(x,y)=(-8y,2x)\ .$$
We can still multiply this ${\bf v}_*$ by an arbitrary nonzero function $\rho(x,y)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1664307",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Is the integral of square of a function (with parameter) positive? Suppose we have a function $f:\mathbb{R}\times\mathbb{R}_{>0}\mapsto\mathbb{R}$, with $f(x;p)\not\equiv 0$, where $p$ is some parameter. Supposing the integral is finite, I know that
$$\int_{\mathbb{R}}f(x;p)^2dx\equiv f_1(p)>0.$$
Does the inequality pe... | It s definitely true. It is the monotonicity property of integrals. If you have two functions $f$ and $g$ such that $f\ge g$, then $$\int f d\mu\ge\int g d\mu.$$
Moreover, if $f>g$ in a set of positive measure, then the inequality above is also strict. Have a look at this too.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1664438",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Imposing non-negativity constraint on a linear regression function Suppose I am interested in estimating the linear regression model
$$
Y_i = g(X_i)^T\beta + \epsilon_i
$$
where $Y_i$ is a scalar outcome of interest, $X_i$ is a scalar covariate with support on the unit interval, $g(\cdot)$ is a $K$-dimensional vector o... | Assuming $g$ is polynomial, one way to attack the functional constraint on non-negativity is to employ a sum-of-squares approach. A sum-of-squares approach (conservatively) replaces a non-negativity constraint $g(x)\geq 0$ with a condition that the polynomial is a sum of squares. When non-negativity only is required on... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1664546",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Prove these functions $F_n$ are bounded by a function $G$
Hi,
I know that to apply the Dominated Convergence Theorem to these functions, they must be bounded by another function $G$ which does not depend on $n$, for all $x$.
However I'm really struggling to find a function.
In the first case, $F_n$ behaves differently... | For second, note that
$$
\lim_{n\to\infty}\int_0^{n^2}xe^{-x^2}dx=\lim_{n\to\infty}\frac{1-e^{-n^4}}{2}=\frac1{2}
$$
Since the Taylor series of $\sin x$ is alternating, for $x>0$ we have
$$
|\sin x-x|\leqslant \frac{x^3}{6}
$$
Thus
\begin{align}
\left|\int_0^{n^2}e^{-x^2}n\sin{\frac{x}{n}}dx-\int_0^{n^2}xe^{-x^2}dx\rig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1664692",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Find out if the transformation is linear, if so, determine whether it is an isomorphism.
Hi, I know how to show that this is a linear transformation. But I am not sure how to figure out if it isomorphic. I tried performing the transformation with M = (a, b, c, d).
When I multiply this out I get a matrix (-2a, -3b, -c,... | Hint: consider $M $ a generic matrix $2×2$ and show that $\ker T=0$ then $M=0$
Edit:
Consider $$M=
\begin{matrix}
a & b \\
c & d \\
\end{matrix}
$$
To find $\ker T$ $T(M)= null matrix$
If find that $a=b=c=d=0$ $M$ is the null matrix and the $\ker $ is zero
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1664909",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Functional calculus for unitization of an algebra? I have been stuck on this problem for a week now:
"Let $A$ be a Banach algebra without identity, let $a\in A$, and let $f$ be holomorphic on a neighborhood of $\sigma(a)$, so that $f(a)\in A^\# $ is defined. Show that $f(a)\in A$ if and only if $f(0) = 0$."
Here $A^\#$... | Hint 1: If $f(0) \ne 0$, consider
$$
f(0) = \frac{f(0)-f(z)}{z}z+f(z)
$$
Hint 2: If $\lambda \in\rho(a)$ in $A^{\sharp}$, then $a(a-\lambda e)^{-1}\in A$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1664997",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
What do the square brackets mean in $[5-(6-7(2-6)+2)]+4$? While watching a youtube video about a Simpsons math episode at 1:27 there's a puzzle that includes square brackets.
$$[5-(6-7(2-6)+2)]+4$$
Apparently the answer is $-27$ which I can't figure out how to arrive at that answer. I've Googled that the square bracket... | $$[5-(6-7(-4)+2)]+4 \implies [5-(6+28+2)] + 4 \implies [5-36] +4 \implies -27$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1665212",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 7,
"answer_id": 5
} |
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