Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
$\sum^n_{i = n+1} 1$ is $0$ or underfined or identity element for the operation it's in? Let's say you have the $\sum^n_{i = n+1} 1$. The things to add don't exist, because $n+1 > n$. What do you do then?
Do you count it as $0$? Because $0$ is the identity element for addition? What if it was multiplying from $n+1$ to ... | An 'empty sum' such as the one you're talking about is defined as 0, because as you say 0 is the additive identity. This is just a convention, though a useful one.
The fact that you then multiply this empty sum by something doesn't change this. The sum still evaluates to 0, which if you multiply by 3 you get 0.
| {
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"timestamp": "2023-03-29T00:00:00",
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A non-nilpotent matrix $A\in \mathbb C^{2 \times2}$ has a square root Is there any quick argument to show that every non-nilpotent matrix $A\in \mathbb C^{2 \times2}$ has a square root? Just the existence without computing it.
Knowing that $A\in \mathbb C^{2 \times2}$ is non-nilpotent basically tells us that at least o... | The result is trivial when $A$ is diagonalisable. So, we only need to consider the case where $A$ is non-diagonalisable and non-nilpotent. Hence $A$ has two equal but nonzero eigenvalues $\lambda$.
By Cayley-Hamilton theorem, $A^2=\operatorname{tr}(A)-\det(A)I=2\lambda A-\lambda^2I$. Therefore $A=\frac1{4\lambda}(A+\la... | {
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Integral: $\int \frac{dx}{\sqrt{x^{2}-x+1}}$ How do I integrate this? $$\int \frac{dx}{\sqrt{x^{2}-x+1}}$$
I tried solving it, and I came up with $\ln\left | \frac{2\sqrt{x^{2}-x+1}+2x-1}{\sqrt{3}} \right |+C$. But the answer key says that the answer should be $\sinh^{-1}\left ( \frac{2x-1}{\sqrt{3}} \right )+C$. Any a... | Notice
$$ x^2 - x + 1 = \left(x-\frac{1}{2}\right)^2 + 1 - \frac{1}{4} = \left(x-\frac{1}{2}\right)^2 + \frac{3}{4}$$
| {
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"timestamp": "2023-03-29T00:00:00",
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Knot theory: Genus of a surface Use Euler characteristic to determine the genus of the surface in Figure 4.24 in picture below.
I am stuck with this question 4.10 from Colin Adams, the Knot Book.
| I know it isn't using the Euler characteristic, but I couldn't help it. Consider the following 'proof by picture':
Start by rounding everything out, so that your original picture looks like a ball with holes drilled out of it. Then just follow the pictures.
This is an ambient isotopy of the manifold, so the genus is ... | {
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A 3-minute algebra problem I have just taken advance math test level 2 and there are several problems that have been bugging me. This is the first question:
If $x,y>0$, then determine the value of $x$ that satisfies the system
of equations: \begin{align} x^2+y^2-xy&=3\\ x^2-y^2+\sqrt{6}y&=3\\
\end{align}
I can answ... | Take G.Bach's hint so that $2y=x+\sqrt 6$
Now add the two equations to obtain $$2x^2+(\sqrt 6-x)y=6$$ Substitute for $y$: $$2x^2+\frac 12(\sqrt 6-x)(\sqrt 6+x)=6=\frac 32 x^2+3$$
So that $x^2=2$
| {
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Compute $\sum_{n=1}^\infty\frac{1}{(1-z^n)(1-z^{n+1})}z^{n-1}$ and show its uniform convergence Given the power series $$P:=\displaystyle\sum_{n=1}^\infty\frac{1}{(1-z^n)(1-z^{n+1})}z^{n-1}$$ I want to show that $P$ converges uniformly in $\mathbb{C}$ and compute its limit.
I've tried to multiply $P$ with $(1-z)z$ whic... | Hint: Note that
$$
\frac{z^{n-1}}{(1-z^n)(1-z^{n+1})}
=\frac1{1-z}\left(\frac{z^{n-1}}{1-z^n}-\frac{z^n}{1-z^{n+1}}\right)
$$
Therefore,
$$
\begin{align}
\sum_{n=1}^N\frac{z^{n-1}}{(1-z^n)(1-z^{n+1})}
&=\frac1{1-z}\left(\frac1{1-z}-\frac{z^N}{1-z^{N+1}}\right)\\
&=1-\frac1{1-z}\frac{z^N}{1-z^{N+1}}
\end{align}
$$
You s... | {
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How to evaluate $\frac{\Gamma\left(\frac{n}{2}\right)}{\Gamma\left(\frac{n-1}{2}\right)}$ How to evaluate $$\frac{\Gamma\left(\frac{n}{2}\right)}{\Gamma\left(\frac{n-1}{2}\right)}$$, where n is integer > 0?
I know the gamma function formula will give
$$ \frac{(\frac{n-2}{2})!}{(\frac{n-3}{2})!}$$ How to simplify it?
| This is an elaboration of the hint by Raymond Manzoni.
One of $n/2$ and $(n-1)/2$ is an integer, and the other is a half integer, and while there is a nice expression of the gamma function on integers as a factorial, evaluating the gamma function on half integers is more complicated. However, they can be evaluated usi... | {
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Distance point on ellipse to centre I'm trying to calculate the distance of a certain point of an ellipse to the centre of that ellipse:
The blue things are known: The lengths of the horizontal major radius and vertical minor radius and the angle of the red line and the x-axis. The red distance is the desired result. ... | We know that the Parametric equation of an Ellipse - not centered, and not parallel to the Axises - is:
$$
x(\alpha) = R_x \cos(\alpha) \cos(\theta) - R_y \sin(\alpha) \sin(\theta) + C_x \\
y(\alpha) = R_x \cos(\alpha) \sin(\theta) + R_y \sin(\alpha) \cos(\theta) + C_y
$$
Where:
- $C_x$ is center X.
- $C_y$ is center Y... | {
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Use Euclid's Algorithm to find the multiplicative inverse Use Euclid's Algorithm to find the multiplicative inverse of $13$ in $\mathbf{Z}_{35}$
Can someone talk me through the steps how to do this? I am really lost on this one.
Thanks
| Basically you want to find $a$ such that $13a \equiv 1 \mod 35$ which is the same as: $$13a + 35k = 1, \qquad \text{For some }k\in \mathbb Z$$
Use Euclid's Algorithm on $13$ and $35$, the same way as for finding $\gcd(13,35)$.
So start with $35 = 2\cdot13 + 9$ and so on... Then substitute your answers in the line above... | {
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Finding volume of the solid of revolution? Can anyone help me with finding the volume of a solid of revolution of f(x) about the x axis for the interval [1,6]. It's supposed to be able to be done without needing calculus but I am having trouble figuring it out.
$f(x) =
\begin{cases}
1 & 1 \leq x< 2\\
1/2 & 2 \leq ... | $f(x)$ is constant in $n$ intervals. Hence, $\int^b_a (f(x))^2dx=(f(x))^2(b-a)$. So, the volume is simply
$$\pi\left(1^2(2-1)+2^2(3-2)+\cdots+\dfrac{1}{n^2}(n+1-n)\right)=\pi\left(1^2+2^2+\cdots+\dfrac{1}{n^2}\right)$$
Though I believe that the $2$ should actually be $\dfrac{1}{2}$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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If $\int_0^1 f(x) e^{nx} dx = 0$ for every n, then f=0 $f$ be a continuous function [0,1] to $R$.
$\int_0^1 f(x)e^{nx} dx = 0$ for all $n \in N\cup\{0\}$
how to prove $f(x)= 0$ in $[0,1]$ for all $x\in[0,1]$?
I solved "$\int_0^1 f(x)x^n dx = 0$ for all $n \in N\cup\{0\}$" with weierstrass theorem
| You haved showed that if $\int_a^b f(x)x^n dx = 0$, then $f=0$. well, we will use it.
Let $e^x=y$,
$$\int_0^1 f(x)e^{nx}\, \mathrm dx=\int_1^e f(\ln y)y^{n-1}\,\mathrm dy$$
so $\int_1^e f(\ln x)x^{n-1}\, \mathrm dx=0$ for all $n \gt 0$. Hence
$$f(\ln x)=0, x\in [1,e]$$
that is $f(x)=0, x\in [0,1]$
| {
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Is the particle in a ring a regular Sturm-Liouville problem? The problem of a particle in a ring is a well-known eigenvalue problem $$\frac{d^2}{d\theta^2} \psi(\theta) + V_0 \psi(\theta) = \lambda \psi(\theta)$$in physics and the Schrödinger equation has a Sturm-Liouville like form. The problem seems to be that the ... | Typical periodic Sturm-Liouville requires two conditions: $\psi(0)=\psi(2\pi)$ and $\psi'(0)=\psi'(2\pi)$.
| {
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Series: Let $S=\sum\limits_{n=1}^\infty a_n$ be an infinite series such that $S_N=4-\frac{2}{N^2}$. Let $S=\sum\limits_{n=1}^\infty a_n$ be an infinite series such that $S_N=4-\frac{2}{N^2}$.
(a) Find a general formula for $a_n$.
(b) Find the sum $\sum\limits_{n=1}^\infty a_n$.
Can you explain to me how I can convert... | Note that $S_{N}-S_{N-1}=a_N$, so that shall give you $a_n$
| {
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Exponential growth precalc population The population of City A increases by 8% every 10 years. The population of City B triples every 120 years. The two cities had equal populations of 10,000 residents each in the year 2000. In what year will city B have twice as many residents as city A?
| Let $t$ the time in years with $t=0$ correspondig to year $2000$, $P_A(t)$ and $P_B(t)$ can be written by
\begin{align}
P_A(t)&=(1.08)^{t/10}P_0& \text{and} & & P_B(t)&=3^{t/120}P_0
\end{align}
where $P_0=10000$, $P_B(t)=2P_A(t)\iff (1.08)^{t/10}=2\cdot3^{t/120}$, taking logs we have
\begin{align}
\frac{t}{120}\log3&=\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Lipschitz and uniform continuity Show that f(x)=$\sqrt{x}$ is uniformly continuous, but not Lipschitz continuous.
I can prove that it's uniformly continuous. But why is it not Lipschits? How do I check the definition?
| Given any $M>0$, choose $c=0$ and $ 0<x<\frac{1}{m^{2}}$, so that $\frac{1}{\sqrt{x}}>M$. then we have:
$$ \frac{|f(x)-f(c)|}{|x-c|}=\frac{|\sqrt{x}|}{|x|}=\frac{1}{\sqrt{x}}>M $$
Since $M$ was arbitrary, this shows that $F$ is not Lipschitz continuous.
| {
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Calculating Eigenvalues is only Assume that the following is used:
$$
A = \begin{pmatrix}
0& 1&\\
2& 3&\\
4& 5&\\
6& 7&\\
8& 9&
\end{pmatrix}
$$
Then calculating the Coveriance matrix, which, gives me:
$$
C = \begin{pmatrix}
40& 40&\\
40& 40&\\
\end{pmatrix}
$$
Then using the following:
$$
det = (a+b)... | Have you double-checked your calculations? The eigenvalues of any $2\times2$ real symmetric matrix $\pmatrix{a&c\\ c&b}$ are given by
$$\frac{a+b\pm\sqrt{(a+b)^2-4(ab-c^2)}}2.$$
Plug in the entries of $C$, I don't find any discrepancies.
| {
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Is this possible physically? The length $x$ of a rectangle is decreasing at the rate of $5 \, cm/min$ and the width $y$ is increasing at the rate of $4\, cm/min$. When $x=8\, cm$ and $y=6 \,cm$, find the rates of change of
(a) the perimeter and (b) the area of rectangle.
$$ \frac{dx}{dt} =-5\,\,cm/min, \, \frac{dy}{d... | So the question is:
Can the perimeter decrease and the area increase?
Absolutely. Consider a rectangle with sides 3 and 4. It has area 12 and perimeter 14.
Now make the short side a bit longer and the long side a bit shorter, we get a new rectangle with sides (for example) 3.1 and 3.89. It has a larger area $$3.1 \cd... | {
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A function such that $f(f(n)) = -n$? This question from somebody's job interview made me puzzled:
Design a function f, such that:
$f(f(n)) = -n$ ,
where n is a 32 bit signed integer; you can't use complex numbers arithmetic. If you can't design such a function for the whole range of numbers, design it for the largest r... | This answer to a related question supplies the relevant analysis.
Assuming that a "signed 32-bit integer" means that you are considering the domain and range of $f$ to be the integers in the interval $[-2^{31}, 2^{31})$, a simple counting argument shows that the largest domain on which the identity $f(f(x))=-x$ can hol... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Solving bernoulli differential equation How to solve $$t \frac{dy}{dt} + y = t^4 y^3$$
First I divided by $t$ to get $$\frac{dy}{dt} + \frac{y}{t} = t^3 y^3$$
Then I multiplied through by $y^{-3}$ to get $$y^{-3} \frac{dy}{dt} + \frac1{ty^2} = t^3$$
Then I used the subsitution $w = y^{-2}$ and $w'=-2y^{-3}\frac{dy}{dt}... | You almost find the solution. You've made a good substitution by letting $w=\frac{1}{y^2}$ and $w'=-\frac{2y'}{y^3}$, then
\begin{align}
\frac{1}{y^3}\frac{dy}{dt}+\frac{1}{ty^2}&=t^3\qquad\rightarrow\qquad-\frac{1}{2}w'+\frac{w}{t}=t^3\qquad\rightarrow\qquad w'-\frac{2}{t}w=-2t^3
\end{align}
The first-order nonlinear ... | {
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Mathematicians ahead of their time? It is said that in every field there’s that person who was years ahead of their time. For instance, Paul Morphy (born 1837) is said to have retired from chess because he found no one to match his technique that very much resembled modern chess theory.
So, who was the Paul Morphy of m... | Simon Stevin (1548 – 1620) had the unbelievable insight of thinking of an arbitrary number in terms of its unending decimal expansion. He was ahead of his time in the sense that the full significance of what he did was not appreciated until the 1870s when more abstract versions of the construction of the reals were giv... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "102",
"answer_count": 27,
"answer_id": 21
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Determine if $\displaystyle\sum_{k=1}^\infty\sin(kx)\sin\left(\cfrac{1}{kx}\right)$ is convergent Determine if the following series converges absolutely:
$$\sum_{k=1}^\infty \sin(kx)$$
and
$$\sum_{k=1}^\infty\sin(kx)\sin\left(\cfrac{1}{kx}\right)$$
I know how to deal with whether they converge. First one diverge by n-t... | The series
$$\tag1 \sum_{k=1}^\infty \sin(kx)$$
converges absolutely for $x\in\pi \mathbb Z$. For all other cases, it doesn' even converge:
Note that
$$ \sin((k+1)x)=\sin(kx)\cos x+\cos(kx)\sin x$$
hence whenever $\sin(kx)\approx 0$ and hence $|\cos(kx)|\approx1$, then $|\sin((k+1)x)|\approx|\sin x|$.
More precisely... | {
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Understanding why $f(z) \cdot (z - p_1)^{n_1} \cdot \ldots \cdot (z - p_k)^{n_k}$ is analytic under a condition Setting: Let $f$ be meromorphic on $\mathbb{\hat{C}} = \mathbb{C} \cup \{\infty\}$. Let $\{p_i\}$ be the $k$ number of poles of $f$. Let $n_i$ denote the orders of each of the $p_i$.
Question: Why is it t... | If $f$ had a pole $z_1$ with order $a$. Then $g(z)=f(z)(z-z_1)^a$ is continuous in $z_1$ and holomorphic in a region of $z_1$. You can see then that it is holomorphic to this region .
| {
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"timestamp": "2023-03-29T00:00:00",
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Conditions on certain entries of a matrix to ensure one Jordan block per eigenvalue In preparation for a future exam, I found the following problem:
Let $$A = \begin{pmatrix} 1 & 0 & a & b \\ 0 & 1 & 0 & 0 \\ 0 & c & 3 & -2 \\ 0 & d & 2 & -1 \end{pmatrix}$$ Determine conditions on $a,b,c,d$ so that there is only one J... | Another approach: There is only one Jordan Block for each eigenvalue of a matrix when the minimum polynomial is the same as the characteristic polynomial. You have the characteristic equation: $(x-1)^4$, so you just need the minimum polynomial to be the same. This is equivalent to $(A-I)^3 \neq 0$. Now \begin{equation}... | {
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"timestamp": "2023-03-29T00:00:00",
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Solutions to differential equation Let $\left\{a,\lambda\right\}\subset\mathbb{R}$.
Let the following differential equation for a function $x\left(t\right)\in\mathbb{R}^{\mathbb{R}}$ be given:
$$ \boxed {\ddot{x}\left(t\right)=4\lambda\left(x\left(t\right)^{2}-a^{2}\right)x\left(t\right) } $$
I am trying to find all so... | These are indeed all solutions. Multiplying the equation by $2\dot{x}$, one easily finds a first integral (the energy):
$$\dot{x}^2=2\lambda \left(x^2-a^2\right)^2+E,\tag{1}$$
and then boundary conditions imply $E=0$.
Now write $x=a\tanh u$, then (1) with $E=0$ gives the equation $\dot{u}^2=2\lambda a^2$, with the only... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/773208",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Convergence/Divergence of $\sum_{n=2}^{\infty} \frac{2^n}{3^n+4^n}$ Using Comparison Test Use the comparison test to show if the series converges/diverges?
$\sum_{n=2}^{\infty} \frac{2^n}{3^n+4^n}$
| Hint:
$\dfrac{2^n}{3^n+4^n}\leq\dfrac{2^n}{3^n}$
Does $\dfrac{2^n}{3^n}$ converge?
| {
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Why are roots (x-intercept) called the solutions of a quadratic equation? As I understand it, all the points in the hyperbole are the solutions. Although I see that it's easier to draw the curve after you discovered the roots, I don't see why to call them 'the solutions.' It has infinite solutions.
| It is by definition. The x's that make the entire equation equal 0 are called the "solutions" to the equation. In other words, the x coordinates of the points where the graph intersects the x axis would be the solutions or otherwise called zeros.
| {
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Tricky Triangle Area Problem This was from a recent math competition that I was in. So, a triangle has sides $2$ , $5$, and $\sqrt{33}$. How can I derive the area? I can't use a calculator, and (the form of) Heron's formula (that I had memorized) is impossible with the$\sqrt{33}$ in it. How could I have done this? The ... | From the law of cosines ($C^2 = A^2 + B^2 - 2AB\cos \theta$), we get that $(\sqrt{33})^2 = 2^2 + 5^2 - 2 \cdot 2 \cdot 5 \cos \theta$.
Simplifying this, we get $33 = 29 - 20 \cos \theta$, which means that $\displaystyle \cos \theta = -\frac{1}{5}$
Because $\cos^2 \theta + \sin^2 \theta = 1$, we get that $\displaystyle ... | {
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Point separating function I'm trying to prove that the set $P = \{p: [0,1] \times [0,1] \to R \; \mid \; \text{p is a polynomial}\}$ is dense in $C( [0,1] \times [0,1], R)$. I'm stuck trying to find a points separating function. (obviously using the Stone Wierestrass approx).
Here is what I got so far: I thing the fun... | I'm sorry I misunderstood your notation initially.
$x \mapsto \sin(x)$ isn't a polynomial function, so we'll have to try something else.
Here's something. Pick a point $(x_0,y_0) \in [0,1]^2$. Then the square of the distance between $(x_0,y_0)$ and another point $(x,y)$ is a familiar polynomial in $x$ and $y$, and is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/773586",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why a linear numerator for fractions with irreducible denominators? For example: (2x^3+5x+1)/((x^2+4)(x^2+x+2)) breaks down to (ax+b/(x^2+4))+(cx+d/(x^2+x+2)). I have been told that since the denominators are irreducible, the numerators will be either linear or constant. Now my question is for something like (2x^3+5x+1... | With a fraction like $ax+b\over cx+d$ it is always possible to remove $a\over c$ copies of the denominator from the numerator, leaving a constant in the numerator:
$${ax+b\over cx+d}={ax+b-ax-\frac ac d\over cx+d}+\frac ac={b-\frac{ad}c\over cx+d}+\frac ac$$
In other words, we have the choice of leaving the fraction in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/773661",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to prove that this function is continuous? If $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ is continuous on the rectangle $R=[a,b] \times [c,d]$, prove that the function
$g(x) := \int\limits_{c}^{d} f(x,y) dy$
is continuous on $[a,b]$.
Thanks in advance!
| Here is how you advance. We need to prove that
$\forall \epsilon >0 $ thers exists $\delta$ such that
$$ |h|<\delta \implies |g(x+h) - g(x)| < \epsilon $$
We advance as
$$ g(x+h) - g(x)= \int\limits_{c}^{d} f(x+h,y) dy - \int\limits_{c}^{d} f(x,y) dy $$
$$ = \int\limits_{c}^{d} (f(x+h,y) - f(x,y)) dy $$
$$ \implies... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/773740",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why is max($\frac{2}{||w||}$)= min($\frac{1}{2}$)($||w||^2$)? I was watching a video on machine learning.
The instructor says that maximizing ($\frac{2}{||w||}$)is difficult (why?) so instead we prefer to minimize $\frac{1}{2}||w||^2$. $w$ is a vector.
How are these two functions equivalent?
****ADDENDUM ****
I plotted... | Let $x = ||w||$, then $\dfrac{2}{||w||} = \dfrac{2}{x}$. So if $\dfrac{2}{x} \leq M$, then:$x \geq \dfrac{2}{M}$, and it follows that: $x^2 \geq \dfrac{4}{M^2}$. This means that instead of finding the max of $\dfrac{2}{x}$ which can be cumbersome, you equivalently solve for the min of $x^2$ which is much simpler becaus... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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The sign representation of the Symmetric Group I am currently trying to learn some of the basics of Representation Theory through Sagan's The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions.
On page 11, after briefly describing the concept of a submodule, he provides an example with ... | It suffices to prove that $$\sigma\cdot w=sgn(\sigma)w$$ for all $w\in W$. Indeed, in this case, any vector space isomorphism $W\rightarrow\mathbb{C}$ can be seen to be an isomorphism of $W$ with the sign representation.
Note that $$\sigma\left(\sum_{\pi\in S_n}sgn(\pi)\pi\right)=\sum_{\pi\in S_n}sgn(\pi)\sigma\pi$$ $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/773917",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Trigonometric identity involving sum of "Dirichlet kernel like" fractions Computing the eigenvalues of a matrix related to fast Fourier transform, we stumbled upon the following identities.
Let $k$ and $N$ be natural numbers with $k<N$, then:
$$\sum\limits_{j=1}^N (-1)^{j+1}\frac
{\sin \left( \frac{2\pi j k}{2N+1}\rig... | (In the spirit of Dirichlet kernel &c) let's rewrite the sum as
$$
\sum_{j=1}^N(-1)^{j+1}2\left[\cos\left(\frac{\pi j}{2N+1}\right)+\cos\left(\frac{3\pi j}{2N+1}\right)+\dots+\cos\left(\frac{(2k+1)\pi j}{2N+1}\right)\right]
$$
and change the order of summation — since
$$
2\sum_{j=1}^N(-1)^{j+1}\cos\left(\frac{(2l+1)\pi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/774045",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Calculate $ \frac{\partial f}{\partial x} (x,y) $ Calculate $ \frac{\partial f}{\partial x} (x,y) $ of
$$ f(x,y) = \int_{x^2}^{y^2} e^{-t^2}\, dt$$
| Let us call $E=E(t)$ a fixed antiderivative of $e^{-t^2}$. Then $$f(x,y)=E(y^2)-E(x^2).$$ Now, $$\frac{\partial f}{\partial x} = \frac{\partial}{\partial x} \left( E(y^2)-E(x^2) \right) = -2x E'(x^2)=-2x e^{-x^4}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/774144",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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$n$ is a square and a cube $a^2 = n = b^3\Rightarrow n\equiv 0,1\pmod{7}$
Verify that if an integer is simultaneously a square and a cube, then it must be either of the form ${7k}$ or ${7k +1}$.
I have no idea on how to proceed.
| If an integer $a$ is simultaneously a square and a cube, it must be the sixth power of an integer $b$
Now as $7$ is prime either $7|b$ or $(7,b)=1$
If $7|b, 7|b^n$ for integer $n\ge1$
Else by Fermat's Little Theorem , $7|(b^{7-1}-1)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/774237",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
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Factorising $D^{2n} - I$ and $D^n - I$ $\rm(ii)$ Let $\mathcal P_{11}(\Bbb R)$ be the vector space of polynomials of degree $\leqslant11$ over the field $\Bbb R$ and let $D:\mathcal P_{11}(\Bbb R)\to\mathcal P_{11}(\Bbb R)$ be the linear map given by differentiation. Write down the least positive integer $n$ for which ... | Hint: Use polynomial division.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$\mathbb{C}\otimes_\mathbb{C} \mathbb{C} \cong \mathbb{R}\otimes _\mathbb{R} \mathbb{C}$ I am trying to show that $\mathbb{C}\otimes_\mathbb{C} \mathbb{C} \cong \mathbb{R}\otimes _\mathbb{R} \mathbb{C}$ as abelian groups.
I've tried to come up with various maps but gotten nowhere.
Thanks for any help
| The groups $\mathbb C\otimes_{\mathbb C}\mathbb C$ and $\mathbb R\otimes_{\mathbb R}\mathbb C$ are both isomorphic to $\mathbb C$. More general, for every $\mathbb C$-vector space $V$ you have $\mathbb C\otimes_{\mathbb C}V\cong V\cong\mathbb R\otimes_{\mathbb R}V$ because for any unitary (commutative) ring $A$ and any... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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1-dim. linear finite elements, von neumann boundary condition. We have the inhomogeneous poisson equation $-\Delta u = f$ on the unit interval and a non-uniform mesh $0=x_0 \ldots x_N=1$. When we write down the stiffness matrix we get a linear system of dimension $N+1$.
Enforcing Dirichlet boundary conditions, the dime... | A Neumann boundary condition simply gets into the system of equations through the Green's first identity when applied to the Laplacian term, i.e.:
$$ \int_L u_{xx} \, \omega \, dx = u_x(1) \omega(1) - u_x(0) \omega(0) - \int_L u_x \omega_x \, dx, $$ where $\omega$ is the test function used in your FEM approach. Since ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/774803",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Real Analysis convergence help I want to show that $\sum \frac{x_{k}}{(1+ x_{k})}$ diverges if $\sum x_{k}$ diverges. I am asked to consider separately where the sequence is bounded and when it is not.
For the bounded case I believe that I have been able to show converges by the Comparison Test.
However, I am stuck at... | In the unbounded case, $x_k/(1+x_k)$ becomes close to $1$ and hence bounded away from zero, for infinitely many $k$. Thus the series cannot converge because the terms do not go to zero. The hard case is when $\{x_k\}$ is bounded.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/774905",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Finding the Polar Equation of a line
Hello, this problem is giving me some trouble. I'm confused as to how I find the equation of the line. I know that x^2+y^2 = r^2, x=r$\cos\theta$, y=r$\sin\theta$, and that tan$\theta$= y/x. I'm just unsure as to how I can derive the polar equation.
| The equation of the circle is $$x^2+y^2=370$$ Differentiating implicitly, we find $$2x+2y\frac{dy}{dx}=0$$ so $$\frac{dy}{dx}=-\frac{x}{y}$$So the slope of your line is $-\frac{19}{3}$. Therefore, in Cartesian coordinates, the slope of the line is $$y=-\frac{19}{3}(x+19)-3$$Now substitute $x=r\cos\theta$ and $y=r\sin\t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/774979",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Limit as x approches infinity - Trouble with calculus I have this problem on a practice exam:
$\displaystyle \lim_{x\to\infty} 3x - \sqrt{9x^2+2x+1}$
We are dealing with L'hospitals rule, so when you plug $\infty$ in for $x$ you get
$\infty - \infty$. I multiplied by the conjugate to get:
$\displaystyle \frac{-2x-1}{3... | Let $L$ be the given expression, then $L = \dfrac{-2x - 1}{3x + \sqrt{9x^2 + 2x + 1}} = \dfrac{-2 -\dfrac{1}{x}}{3 + \sqrt{9 + \dfrac{2}{x} + \dfrac{1}{x^2}}} \to \dfrac{-2}{6} = \dfrac{-1}{3}$ when $n \to \infty$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/775073",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Difficult time finding critical points using Lagrange The function is $f(x,y,z) = xyz$ on $x^2 + y^2 + z^2 = 1$. So I have:
$yz = 2x \lambda \\
xz = 2y \lambda \\
xy = 2z \lambda \\
x^2 + y^2 + z^2 = 1$
I guessed $x = \pm 1, y = 0, z = 0, \lambda = 0$ but apparently this isn't a critical point. There's many more that I... | Perhaps something like this:
$$ yz = 2x \lambda |\times x\\
xz = 2y \lambda |\times y \\
xy = 2z \lambda |\times z\\ $$
$$ xyz = 2x^2 \lambda\\
xyz = 2y^2 \lambda \\
xyz = 2z^2 \lambda \\ $$
Now we can something like this for example divide firs two equations and get:
$$ x^2=y^2=z^2 $$
Now solve this step by step - ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/775184",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to find the values of a and b? If the polynomial 6x4 + 8x3 - 5x2 + ax + b is exactly divisible by the polynomial 2x2 - 5, then find the values of a and b.
| The roots of $2x^2-5$ must be roots of our polynomial:
$$\left\lbrace
\begin{array}{c}
6\cdot\frac{25}4+15\sqrt{\frac52}-\frac{25}2+\sqrt{\frac52}a+b=0\\
\phantom{1}\\
6\cdot\frac{25}4-15\sqrt{\frac52}-\frac{25}2-\sqrt{\frac52}a+b=0
\end{array}
\right.$$
Adding up the equations you get
$$50+2b=0$$
And substracting them... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Quadratic Equality Constraints via SDP I want to know if it is possible to solve a QCQP problem with quadratic equality constraints in SDP. I know it is possible to convert a QCQP to an SDP by using the Shur complement. The following worked for me thus far:
$$
\begin{equation}
\begin{array}{cccccc}
\underset{x}{min} & ... | Hint:
You don't need the constraints $M_{j}M_{j}^{T}=Q_{j}$. For example if $UDU^T$ is the eigen decomposition of $Q_j$, Then $M_j=UD^{\frac{1}{2}}$ (if $Q_j$ is symmetric)
Note that if you introduce new variable $M_j$, then the bilinear forms in the constraints make your problem non-convex, (Besides the fact that equa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/775343",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Calculate $\lim_{n\rightarrow\infty}\frac{1}{n}\left(\prod_{k=1}^{n}\left(n+3k-1\right)\right)^{\frac{1}{n}}$ I'm need of some assistance regarding a homework question:
$$
\mbox{"Calculate the following:}\quad \lim_{n \to \infty}
\frac{1}{n}\left[%
\prod_{k = 1}^{n}\left(n + 3k -1\right)\right]^{1/n}\
\mbox{"}
$$
Alrig... | The product $P_n$ may be expressed as follows:
$$P_n = \left [ \prod_{k=1}^n \left (1+\frac{3 k-1}{n}\right ) \right ]^{1/n} $$
so that
$$\log{P_n} = \frac1{n} \sum_{k=1}^n \log{\left (1+\frac{3 k-1}{n}\right )}$$
as $n \to \infty$, $P_n \to P$ and we have
$$\log{P} = \lim_{n \to \infty} \frac1{n} \sum_{k=1}^n \log{\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/775439",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How come if $\ i\ $ not of the following form, then $12i + 5$ must be prime? I know if $\ i\ $ of the following form $\ 3x^2 + (6y-3)x - y\ $ or $\ 3x^2 + (6y-3)x + y - 1, \ \ x,y \in \mathbb{Z}^{+},i \in \mathbb Z_{\ge 0}$,
then $\ 12i + 5\ $ must be composite number, because:
$12(3x^2 + (6y-3)x + y - 1) + 5 = 36x^2 ... | This is rather elementary.
Suppose that $12i+5$ is composite, say $12i+5 = ab$.
Then looking modulo $6$, we get that one of the elements $a,b$ is $1 \pmod{6}$ and the other is $-1 \pmod{6}$; assume w.l.o.g. that $a$ is $1 \pmod{6}$, and write $a = 6r+1$ and $b = 6s-1$.
Observe that $r$ and $s$ must have different parit... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Topology of space of symmetric matrices with fixed number of positive and negative eigenvalues Let $M$ be real non-singular symmetric $n \times n$ matrix with $p$ positive and $n-p$ negative eigenvalues. What is the topology of the space of such matrices?
For a trivial case $n=1$ the matrix is an ordinary number (which... | Here are my comments as an answer (with few more details):
Suppose you fix the eigenvalues and set $q=n−p$. Then, in view of the orthogonal diagonalization of quadratic forms, what you get is the homogeneous space $$O(n)/(O(n)\cap O(p,q))=O(n)/(O(p)\times O(q))$$
which is the Grassmannian $G_p(R^n)$ of $p$-dimensional ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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If $a_n$ is a strictly increasing unbounded sequence, does $\sum_n \frac{a_{n+1} - a_n}{a_n}$ diverge? So I've been thinking through some test cases. If $a_n = n$ then $\sum_n \frac{a_{n+1} - a_n}{a_n}$ is the harmonic series which diverges. And if $a_n = \sum_{k=1}^n 1/k$ then $\sum_n \frac{a_{n+1} - a_n}{a_n}$ diverg... | Yes.
Let $b_n=\frac{a_{n+1}-a_n}{a_n}> 0$. Suppose to the contrary that $\sum_n b_n<A$. We show that
$$
a_{n+1}=a_n (1+b_n)=a_1\Pi_{1\le k\le n}(1+b_k)
$$
converges, contradicting the premise.
In fact
$$
\Pi_{1\le k\le n}(1+b_k)\le (\frac{\sum 1+b_k}{n})^n\le (1+\frac{A}{n})^n\to e^A.
$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is there a generating function for $\sqrt{n}$? I tried to come up with a closed form for the ordinary generating function for the sequence $\{\sqrt{n}\}_0^{\infty}$ but I could not. Is there a way to derive it using the recurrence relation $$a_{n+1} = \sqrt{a_n^2+1}. $$
Because if there is, it is no obvious to me how t... | It does exist, defined as $g(z) = \sum_{n \ge 0} \sqrt{n} z^n$. It is even a nice function, in that it is analytic in a region around the origin (apply your favorite test). It doesn't have a representation in terms of elementary functions, however.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/776007",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
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is 121 divides the my pattern for base 2? Is $121|2^{120}-1$? If yes, is there any online free calculation to check these type of values?
Advanced thanks to one and all!
-Richard Sieman
| If $p$ is prime, then you have $p$ divides $2^{p-1} - 1$ by Fermat's little theorem, which would show that your result holds if $121$ is prime. However you have that $121 = 11^2$ is not prime, so it's more complicated. A simple calculation shows that the congruence indeed does not hold.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/776079",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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A closed form for the infinite series $\sum_{n=1}^\infty (-1)^{n+1}\arctan \left( \frac 1 n \right)$ It is known that $$\sum_{n=1}^{\infty} \arctan \left(\frac{1}{n^{2}} \right) = \frac{\pi}{4}-\tan^{-1}\left(\frac{\tanh(\frac{\pi}{\sqrt{2}})}{\tan(\frac{\pi}{\sqrt{2}})}\right). $$
Can we also find a closed form for th... | This is just a possible way to proceed, and doesn't provide in any way a complete solution to the problem, thus I will put it as community wiki, and pray you to contribute if you have any insights.
The series converges (check!), inserting the Taylor series for $\arctan(x)$:
$$\begin{array}{ll}\sum_{n=1}^\infty(-1)^{n+1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/776182",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 6,
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$r=(x,y,z)$ prove that $\mathrm{curl}\; r = 0$
Example $\bf 84\,\,\,$ Let $\,\mathbf r=(x,y,z)$ and $r=|\!\,\mathbf r|=\sqrt{x^2+y^2+z^2}$. Then $$\operatorname{div}\mathbf r=
\dfrac{\partial x}{\partial x}
+
\dfrac{\partial y}{\partial y}
+
\dfrac{\partial z}{\partial z}
=3; \\
\operatorname{curl}\mathbf r=
\left|\be... | The reason the curl is $0$ is because $\mathbf{r}$ has continuous second-order partial derivatives. It's a known theorem.
You should also note that this immediately implies $\mathbf{r}$ is a conservative field.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/776256",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Does a series have to start from 1 to be able to apply convergence tests? In the ratio test, for example, we know that the test applies for a sum from n=1 to n=infinity.
Can I directly apply this test to sums from n=x to n=infinity (for example, a sum from n=0 -> infinity)?
| You can always write your sum as a sum starting with the index $1$ by performing an index shift:
$$
\sum_{k=k_0}^\infty a_k = \sum_{k=1}^\infty a_{k_0+k-1}.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/776353",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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On the hypothesis of the Additive Cousin Problem The Additive Cousin Problem is the following:
Assume that $D$ is a region (open, connected) of $\mathbb{C}^n$. Assume that the Dolbeault Cohomology Group of $D$, $H^1_{\bar{\partial}}(D)$ is equal to zero, and that we have an open covering of $D$ given by $\{ U_j:j\in J... | No, the problem really assumes you have meromorphic functions satisfying certain conditions (perhaps with certain poles). You could always take all the $m_j$ to be constant and then there's no hypotheses at all needed for them to exist.
Perhaps you already understand how this goes, but if $H^{0,1}_{\bar\partial}(D)=0$... | {
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is there a higher dimensional analogue of the first isogonic center? I'm curious to know if, given four points $a, b, c, d$, you can always find a point $p$ such that last lines $pa, pb, pc, pd$ form equal angles pairwise.
I'd also appreciate resources on 3d geometry especially if there is an analogue of inscribed ang... | No, there is not. Take $ABC$ and $BCD$ as two equilateral triangles, let $M$ be the midpoint of $BC$, $\pi_A,\pi_D,\pi_M$ be the planes where $ABC,BCD$ and $ADM$ lie. The locus $l_A$ of points $P$ such that $P$ "see" all the sides of $ABC$ under the same angle is a line through the center of $ABC$ orthogonal to $\pi_A$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/776516",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to find a counter example for non convexity? Consider a simple function $f(x,y)=\frac{x}{y}, x,y \in (0,1]$, the Hessian is not positive semi definite and hence it is a non convex function. However, when we plot the function using Matlab/Maxima, it "appears" convex. For the sake of clarity we want to find points wh... | If you write down the Hessian matrix you'll see that for $x,y\in (0,1]$ the Hessian is indeed positive semi-definite. This is call conditional positive semi-definiteness. You can see the Hessian as a matrix parametrized by $x$ and $y$: it is positive semi-definite depending on their values. Try to compute the eigen-val... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/776615",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Behaviour of Solutions to $x^2y'' + \alpha xy'+ \beta y = 0$ as $x \to 0$ and $x \to \infty$
Consider the Euler equation $x^2y'' + \alpha xy' + \beta y = 0$.
Find conditions on $\alpha$ and $\beta$ so that:
*
*All solutions approach zero as $x \rightarrow 0$
*All solutions are bounded as $x \rightarrow ... | A logarithmic term, if present, only changes the behaviour as $x \to 0$ or $x \to \infty$ when $\text{Re}(r) = 0$. That is, if $\text{Re}(r) \ne 0$ then both
$x^r$ and $x^r \log(x)$ have the same limit or lack of limit as $x \to \infty$ and as $x \to 0$. I don't know what that "or complex" is doing there: in the Eul... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/776696",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Are rank of $T$ and $T^*$ equal? Let $H$ be a infinite dimensional Hilbert space and $T:H\to H$ be an operator on $H$.Is it true that $\operatorname{rank}T=\operatorname{rank}T^*$. We know that this is true for finite dimension. Is it true for infinite dimensional Hilbert space or is there any example of $T$ and $T^*$ ... | The operator $Tv \mapsto T^*v$ is actually a partial isometry. The closure of the ranges are Murray-von Neumann equivalent.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/776813",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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A subsequence of a convergent sequence converges to the same limit. Questions on proof. (Abbott p 57 2.5.1) Solutions to Homework 3 doesn`t duplicate. We have to prove that if $(a_{n})$ is a sequence in $\mathbb{R}$ with $\displaystyle \lim_{n\rightarrow\infty} a_n =a$, and if $(a_{n_{k}})_{k\in \mathbb{N}+}$ is a subs... | I'm going to post a separate answer because I think I can be clearer than the accepted answer.
I know we must find $ N\in\mathbb{N}$ such that $\color{red}{n_k} \ge N\implies |a_\color{red}{n_k}-a| < e \quad (♫)$.
No, this is not what you need to show! You need to show that there exists $N$, such that for all $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/776880",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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How to solve this limit, hint only $$\lim_{n\to\infty}\bigg(\frac{1}{\sqrt{9n^2-1^2}}+\frac{1}{\sqrt{9n^2-2^2}}+ \dots +\frac{1}{\sqrt{9n^2-n^2}}\bigg)$$ I need a hint. I see that maybe compute with integral. But what the integrable function?
| $$\text{As }\lim_{n \to \infty} \frac1n\sum_{r=1}^n f\left(\frac rn\right)=\int_0^1f(x)dx$$
Here the $r$( where $1\le r\le n$)th term $=\displaystyle\frac n{\sqrt{9n^2-r^2}}=\frac1{\sqrt{9-\left(\frac rn\right)^2}}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/776960",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Orthogonal Projector Let $(H,\langle\,|\,\rangle)$ be a separable Hilbert space on $\mathbb{C}$.
$P_{\psi}:=\langle\psi,\,\rangle\psi$, where $\psi\in H$ is such that $\|\psi\|=1$.
I have to prove that $P_{\psi}$ is an orthogonal projector on $H$.
I can prove $P_{\psi}P_{\psi}=P_{\psi}$, but I can't prove $P_{\psi}^*... | For $\xi,\eta\in H$ we have
$$\langle P_\psi(\xi),\,\eta\rangle\ =\ \big\langle \langle \psi,\xi\rangle\cdot\psi\,,\ \eta\big\rangle\ =\ \langle\psi,\xi\rangle\cdot\langle \psi,\eta\rangle$$
(assumed that conjugation happens in second variable).
And, we get the same for $\langle \xi,\,P_\psi(\eta)\rangle\,$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/777042",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Green's Function Solutions So, I'm considering a PDE and trying to find its Green's function first. To this end, I solve the following helmholtz equation:
$$\frac{d^2g}{dx^2}+\frac{d^2g}{dy^2}+\frac{d^2g}{dz^2}-\alpha^2g=\delta(x-\xi)\delta{(y-\eta)}\delta{(z-\rho)}$$
Well, I can solve this PDE for $g$ , but what hap... | Yes, the Green's function (a.k.a. "fundamental solution") does not solve the homogeneous equation "at" the separation point.
There are at least two ways to think about this. One is to approximate (weakly) the Dirac deltas by spikes that are nevertheless smooth pointwise-valued functions. For elliptic operators and such... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/777126",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How can I define $e^x$ as the value of infinite series? I understand the definition of $e^x$ by limit. But I do not know how to come up with:
$$e^x = \sum_{n=0}^{\infty}\frac{x^n}{n!}$$
without using Taylor series. more explicitly without using calculus. how do we know if a function can be expressed as series or not ?
| One definition of the exponential function is the limit
$$
\lim_{n\to\infty} \Big(1 + \frac{x}{n}\Big)^n=e^x.
$$
Let $P_n(x)$ denote the polynomial $(1+x/n)^n$, so that $e^x=\lim_{n\to\infty} P_n(x)$; I will show that
$$
\lim_{n\to\infty} P_n(x) = \sum_{n=0}^\infty \frac{x^n}{n!}.
$$
If you expand out $P_n(x)$ using t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/777224",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Gram-Schmidt process Question:Apply the Gram-Schmidt process to find an orthonormal basis for the subspace.
$S= \mathrm{span} [{(1,2,-4,-1),(-3,0,5,-2),(0,7,2,-6)}]$
The span is suppose to look like a matrix but I couldn't get it to look right here so I wrote it that way.
I was able to do the process but I wanted to ch... | Wow what a nasty question...well I get exactly those $u_1,u_2,u_3$, and I cross-checked on Mathematica - it is correct. I am not sure exactly what you thought should have cancelled out, but it seems as if it is not "wrong" the fact that it didn't cancel out.
But I think it might be worthwhile to try and find a simpler... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Use logarithmic differentiation to find the derivative of $y = (1 +\frac1x)^{2x}$ Can someone guide me through solving this problem?
$$\dfrac{\mathrm d}{\mathrm dx}\left(1 +\dfrac1x\right)^{2x}$$
| Let
$$
y=\left(1 +\dfrac1x\right)^{2x}
$$
then
$$
\ln y=2x\ln\left(1 +\dfrac1x\right).
$$
Thus
$$
\begin{align}
\frac{d}{dx}\ln y&=\frac{d}{dx}2x\ln\left(1 +\frac1x\right)\\
\frac1y\frac{dy}{dx}&=2\ln\left(1 +\frac1x\right)+\frac{2x}{1 +\frac1x}\cdot\left(-\frac1{x^2}\right)\\
&=2\ln\left(1 +\frac1x\right)-\frac{2}{x +... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/777418",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that $\int_0^de^{-tx}g(x)dx\sim_{t\rightarrow\infty} \frac{g(0)}{t}$ Let $d>0$. Let $g\in C^0([0,d])$ such that $g(0)\ne0$.
Show that $$\int_0^de^{-tx}g(x)dx\sim_{t\rightarrow\infty} \frac{g(0)}{t}$$
How can I prove that ?
It's the first time I see this kind of exercise
Thank you in advance for your time
| Here is a start. Making the change of variables $y=tx$ yields
$$ \int_0^de^{-tx}g(x)dx = \frac{1}{t}\int_{0}^{td} g\left(\frac{y}{t}\right)e^{-y} dy. $$
I think you can advance now.
Note:
$$ \int_{0}^{\infty} e^{-y}dy = 1. $$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Prove that if m is prime and m|kl then either m|k or m|l. Proofs homework question, here's what I've figured out thus far.
Suppose m doesn't divide k. We need to then prove that m|l. If m doesn't divide k and m is a prime then we know m and k are co-prime - hcf (m,k) = 1. Which means 1 = ks + mt (for some integers; ... | The Bezout-based proof is a bit more intuitive interpreted modulo $\,m.\,$ Notice that the Bezout identity $\,\color{#c00}{1 = \gcd(k,m)} = a k\! +\! b m\,$ becomes $ $ mod $\,m\!:\ 1\equiv ak,\,$ hence $\,\color{#c00}{k^{-1}} \equiv a$ exists mod $\,m.\,$ Hence $\, m\mid k\,\ell\,\Rightarrow\,k\,\ell\equiv 0\!\!\!\!... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Triple Integral Troubles I'm having trouble calculating this integral. I can do the first one just fine, but it's in simplifying and calculating the third integral where I get stuck.
$16\int_0^\frac{\pi}{4}\int_0^1\int_0^{\sqrt{1-r^2cos^2(\theta)}}rdzdrd\theta$
$16\int_0^\frac{\pi}{4}\int_0^1r\sqrt{1-r^2cos^2(\theta)}... | You're forgetting a "$-1$" term. Let $u = 1 - r^2\cos^2\theta$ so that $du = -2r\cos^2\theta \, dr$. Then observe that:
\begin{align*}
\int_0^1 r \sqrt{1 - r^2\cos^2\theta} \, dr
&= \frac{1}{-2\cos^2\theta}\int_1^{1-\cos^2\theta} \sqrt{u} \, du \\
&= \frac{1}{-2\cos^2\theta}\left[\frac{u^{3/2}}{3/2}\right]_1^{1-\cos^2\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why is calculating the area under a curve required or rather what usage it would provide I understand Integration and Differentiation and see a lot of Physics / Electrical Theory using them.
Take for example a sine wave. So area for me means the space any object would occupy. So what's usage it comes to find the area o... | A function, differentiate it represent how fast it changes, integrate it represent how much change had accumulated.
If we know the derivative of the function (how fast it changes), by integrate it, we find function itself. If we know how much change had accumulated, by differentiate it, we can find of the function of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/777891",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 6,
"answer_id": 5
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a function that is in $L^2$ (the right version) Sorry I made a mistake when posting the last question. Actually my question is: can you give a $f(x) $ such that $ f \in L^2 ( \mathbb R)$ but $ x^{-\frac{1}{2}} f \notin L^1 ( \mathbb R ) $. Thanks!
| Take $$f(x)=\frac 1{\sqrt x\log x}\mathbf{\chi}_{[2,\infty)}(x).$$
Then $x\mapsto f^2(x)=(x\log^2x)^{-1}\mathbf{\chi}_{[2,\infty)}(x)$ is integrable and $x\mapsto x^{-1/2}f(x)=(x\log x)^{-1}\mathbf{\chi}_{[2,\infty)}(x)$ is not.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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$\ln|x|$ vs.$\ln(x)$? When is the $\ln$ antiderivative marked as an absolute value? One of the answers to the problems I'm doing had straight lines:
$$ \ln|y^2-25|$$
versus another problem's just now:
$$ \ln(1+e^r) $$
I know this is probably to do with the absolute value. Is the absolute value marking necessar... | logarithm is a function only defined on domain $(0,\infty)$, so it make no sense to input negative value.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/778068",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "31",
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Show that if $p$ is a prime of the form $p=4n+1$, then we can solve $x^2\equiv -1\mod p$(with $x$ an integer).
Show that if $p$ is a prime of the form $p=4n+1$, then we can solve $x^2\equiv -1\mod p$(with $x$ an integer).
My attempt:If $p$ is a prime, then $U_p=${$[x]|1\leq x<p$} is cyclic.
| The following is an alternative proof that goes back to Dirichlet. Note that for every $x$ in the interval $1$ to $p=1$, there is a unique $y$ in that interval such that $xy\equiv 1\pmod{p}$. Let $p=4k+1$. Suppose that $x^2\equiv -1\pmod{p}$ has no solution. We will show that leads to a contradiction.
If $x^2\equiv -1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/778155",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find the value of the real parameters a,b, if there exists a P(X) binomial and the following is true Find the value of the real parameters a,b, if there exists a P(X) binomial and the following is true
$(X^3-aX^2-bX+1) : P(X) = X^2-X+1 $
I tried to divide and to equalize the remainder to zero, but I think i'm missing s... | you can make a polynomial division like this: $(X^3-aX^2-bX+1):(X^2-X+1)=P(X)$
It´s just a transformation of the origininal equation. Then compare the values of the parameters.
greetings,
calculus.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/778241",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Combinatorics (combinations problem) How many ways are there to pick a group of $4$ people from $10$ people (each of a different height) and then pick a second group of $3$ other people such that all the people in the first group are taller than all the people in the second group? (Hint: Consider two ways)
I try to il... | Just pick $7$ people from $10$, and let the $3$ shortest ones be called the second group. This can be done in $\binom{10}{7}$ ways.
Remark: The cases approach of the post does some double-counting. One can adjust it, by organizing by "shortest in the group of $4$," If the shortest in that group is to be say Person $6$,... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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no. of positive integral solutions of ||x - 1| - 2| + x = 3 What are the no. of positive integral solutions of ||x - 1| - 2| + x = 3 ?
My effort
||x - 1| - 2| = 3 - x
|x - 1| - 2 = 3 - x OR |x - 1| - 2 = x - 3
|x - 1| = 5 - x OR |x - 1| = x - 1
x - 1 = 5 - x OR x - 1 = x - 5 OR x - 1 $\geq$ 0
2x = 6 OR x $\geq$ 1
... | $\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,}
\newcommand{\dd}{{\rm d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Text similar to chapter 9 of Topology from James Munkres I'm self-studying chapter 9 of Topology from James Munkres.
I like to read different books about the same topic at the same time. Can someone recommend some text/book that is about the same subjects as found in chapter 9?
This chapter is about the fundamental gro... | Algebraic Topology by F.H Croom will be a good choice for a beginner in algebraic topology
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/778483",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 3
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key generation in RSA cryptosystem: why it can be performed in polynomial time? Suppose that I want to generate the keys of the RSA cryptosystem: the public key will be the couple $(n,e)$ where $n$ is the product of two primes $p$ and $q$ and gcd$(\phi(n),e)=1$.The private key will be the integer $d$ such that $ed=1$ ... | Step 1 is the hard one! I think that it still hasn't been proven to be polynomial, although it is believed to be.
Step 3 is easy, if all you need is an $e$ such that $(e,\phi(n)) = 1$ (although a real-life public key protocol will require more than this). Just let $e$ be the smallest prime that doesn't divide $\phi(n)$... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Writing answers to trigonometric equation I wonder how to write answers to trigonometric equations in more elegant form. For instance if we have $ \displaystyle \sin x = \frac{\sqrt{2}}{2} \vee \sin x=-\frac{\sqrt{2}}{2}$ then I write four cases instead of just one where $\displaystyle x=\frac{\pi}{4}+\frac{k\pi}{2}$
C... | $$\sin x=-\frac{\sqrt2}2=-\frac1{\sqrt2}=\sin\left(-\frac\pi4\right)$$
$$\implies x=n\pi+(-1)^n\left(-\frac\pi4\right)$$ where $n$ is any integer
for $\displaystyle n=2m\implies x=2m\pi-\frac\pi4$
for $\displaystyle n=2m+1\implies x=(2m+1)\pi+\frac\pi4=2m\pi+\frac{5\pi}4$
Similarly, $\displaystyle\sin x=\frac{\sqrt2}2\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/778691",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Equivalence Relation Proof for modular arithmatic Given this modular relation:
$x^3 \equiv y \pmod{3}$
how would you go about proving the transitivity of the system? I have proven the reflexivity, and symmetry pretty easily but the transitivity is giving me many problems, and I feel like im not setting up the problem c... | By little Fermat (or directly) note $\,{\rm mod}\ 3\!:\ x^3\equiv x\,$ for all integers $\,x,\,$ thus $\,x^3\equiv y\iff x\equiv y.\,$ Thus your relation is the same as the standard congruence relation '$\equiv$', which is transitive since
$\quad x\equiv y,\ y\equiv z\pmod 3\,\Rightarrow\, 3\mid x\!-\!y,y\!-\!x\,\Right... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Find maximum of a complex function $f(z)$ I am trying to find the following maximum, whose existence is justified by the compactness of the close ball $\Delta$ of $\mathbb C$ and continuity of the function
$$f(z)=\sum_{k=1}^\infty (-1)^k \frac{2z}{k^2 \pi^2-z^2}\cos kt$$
from $\mathbb C$ to $\mathbb C$
$$\larg... | Hint: holomorphic functions on some bounded open area attaint their maximum on the boundary of the area.
| {
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"timestamp": "2023-03-29T00:00:00",
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Fibonacci's proof that $x^3+2x^2+10x=20$ has no solution in radicals? I read on a poster today that Fibonacci showed that $x^3+2x^2+10x=20$ has no solution expressible in radicals, way back when.
I couldn't find the proof anywhere. Does anyone know where I can find it?
| He proved that the solution cannot be one of Euclid's irrationals. All Euclid's irrationals are strictly contained in the set of numbers of the form
$$
\sqrt[4]{p}\pm\sqrt[4]{q},
\qquad p,q\in\mathbb{Q}.
$$
The proof would be similar to (but of course more complicated than) how you prove $\sqrt2$ is not rational.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Using random numbers to evaluate $\sum_{k=1}^{10000}e^{\frac{k}{10000}}$ I tried using the Monte Carlo Method to approximate the sum $\sum_{1}^{10000}e^{\frac{k}{10000}}$. First I genarating 100 random numbers in (1, 10000). Then by the strong law of large numbers:
$$\sum_{i=1}^{100}\frac{f(U_i)}{k} \to E[f(U)] = \the... | You are confusing a few different concepts:
*
*You write $\frac{1}{k}\sum_{i=1}^{100}f(U_i)\to\mathbb{E}[f(U)]$ when you clearly mean $\frac{1}{100}\sum_{i=1}^{100}f(U_i)\approx\mathbb{E}[f(U)]$.
*You talk about random numbers in $\{1,\dots,10000\}$ (implying that you choose them uniformly). This is therefore a dis... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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2 Trains and Fly Problem. Find the number of trips made by the fly back and forth. Question: A Train A is approaching at a speed of 10m/sec, another Train B moving in the opposite direction at a speed of 20m/sec. A fly whose absolute speed is 50m/sec goes repeatedly from A to B and back, without loosing any time at any... | The problem can be solved much more simpler by using graphs. A plot of d(t) vs t for the trains A and B will be two straight line who intersect at the time and position where the two trains crash. Furthermore, plotting the d(t) of the bird on the same plane will show clearly where the fly keeps going to and fro between... | {
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"url": "https://math.stackexchange.com/questions/779107",
"timestamp": "2023-03-29T00:00:00",
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"answer_id": 2
} |
Non-Hamiltonian $k$-connected $k$-regular graphs ($k>3$)? The Petersen graph provides us with an example of an non-Hamiltonian 3-connected 3-regular graph. Are any 4-connected 4-regular graphs known to be non-Hamiltonian? What about generic $k$-connected $k$-regular graphs?
| According to this paper (Regular n-Valent n-Connected NonHamiltonian Non-n-Edge-Colorable Graphs by G. H. J. Meredith) the answer is yes (by construction) for $k=4$ and $k \ge 8$ but the other cases are not shown there. You can see a nice picture of the construction (which goes via the Petersen graph) for $k=4$ on page... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/779239",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
$\sum {a_n}$ be a convergent series of complex numbers but let $\sum |{a_n}|$ be divergent.. I am stuck on the following problem that says:
Let $\sum {a_n}$ be a convergent series of complex numbers but let $\sum |{a_n}|$
be divergent.
Then it follows that
a. $a_n \to 0$ but $|{a_n}|$ does not converge to $0$.
b. th... | 1) Is not true. For example, take $a_n = \dfrac{(-1)^n}{n}$, $n > 0$.
2) Also not true. Take the same sequence.
3) Not true. Take $a_n = \dfrac{(-1)^n}{n}$ if $n$ is odd, and $0$ if $n$ is even. Then, $a_n$ converges and $|a_n|$ does not, but there are not finitely many $a_n$'s that are $0$.
4) True. If finitely many $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/779301",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Find a continuous function on the reals where $f(x) >0$ and $f'(x) < 0$ and $f''(x) < 0$ We need to find a function $f(x)$ where $f(x) >0 $and $f'(x) < 0$ and $f''(x) < 0$ where $f$ is continuous for all real numbers.
We have tried $ f(x) = \sqrt{-x}$ however this is not defined for $x>0$ and therefore is only continuo... | Such function does not exist!
If not, there is such $f$ concave, so
$$f(x)\leq f'(0)x +f(0), \qquad \forall x \in \Bbb R$$
Since $f'(0)\lt0$, so if $x \gt -\dfrac{f(0)}{f'(0)}$, then
$$f(x)\leq f'(0)x +f(0)\lt0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/779492",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
easy question about conservative vector fields Suppose I have a vector field $f: A \subset R^2 \to R^2 $. Write $f = (M,N) $. Does it follow this?
$$ \frac{ \partial M}{\partial y} = \frac{ \partial N}{ \partial x } \quad \iff \quad f \text{ is conservative}$$
| No. It does not. If $A$ is the punctured plane then the vector field with $M = -y/(x^2+y^2)$ and $N = x/(x^2+y^2)$ is not conservative. This can be seen by integrating around any closed contour which contains the origin. It will give a nontrivial result depending on how many times the curve winds around the origin. On ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/779562",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Using properties of determinants, show that Using Properties of determinants, show that:
$$
\begin{vmatrix}
a & a+b & a+2b\\
a+2b & a & a+b\\
a+b & a+2b & a
\end{vmatrix}
= 9b^2 (a+b)
$$
I've tried it but not getting $9b^2$
| Add all the three columns to get
$$\left \vert \begin{bmatrix} 3a+3b & a+b & a+2b\\ 3a+3b & a & a+b\\ 3a+3b & a+2b & a\end{bmatrix}\right \vert = 3(a+b)\left \vert \begin{bmatrix} 1 & a+b & a+2b\\ 1 & a & a+b\\ 1 & a+2b & a\end{bmatrix}\right \vert$$
Subtract first row from row and first from third to get
$$3(a+b)\left... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/779673",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Prove the identity $\frac{\cos B}{1-\tan B} + \frac{\sin B}{1-\cot B}=\sin B+\cos B$ I have worked on this identity from both sides of the equation and can't seem to get it to equal the other side no matter what I try.
$\displaystyle\frac{\cos B}{1-\tan B} + \frac{\sin B}{1-\cot B} =\sin B+\cos B$
| HINT:
$\displaystyle\frac{\cos B}{1-\tan B}=\frac{\cos B}{1-\dfrac{\sin B}{\cos B}}=\frac{\cos^2B}{\cos B-\sin B}$
$\displaystyle\frac{\sin B}{1-\cot B}=\frac{\sin B}{1-\dfrac{\cos B}{\sin B}}=\frac{\sin^2B}{\sin B-\cos B}=-\frac{\sin^2B}{\cos B-\sin B}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/779760",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
2 questions about counting and permutation I have a test on monday and I couldn't solve these 2 questions, I'd be grateful if you help me
1-) How many ways are there to distribute 18 balls among 6 dierent persons if
a) each ball is dierent and each person should get 3 balls
b) all balls are identical ?
2-)How many perm... | Question 2: We use the Principle of Inclusion/Exclusion. There are $\frac{12!}{3!3!3!3!}$ permutations (words). We now count the bad words, in which there are $3$ consecutive occurrences of a or of b or of c or of d.
We count the words with $3$ consecutive a. Group the three a into a superletter, which we will call A... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/779859",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Proof using complex numbers Prove that $\left|\dfrac{z-w}{1-\bar{z}w}\right| = 1$ where $\bar{z}$ is conjugate of $z$ and $\bar{z}w\ne 1$ if either $|z| = 1$ or $|w| = 1$.
I used $|c_1/c_2| = |c_1|/|c_2|$ and multiply out with $z = x + iy$ and $ = a+ib$ but I am getting stuck near finish.
| Suppose $|z|=1$: then
$$
\left|\dfrac{z-w}{1-\bar{z}w}\right|=
\frac{1}{|z|}\left|\dfrac{z-w}{1-\bar{z}w}\right|=
\left|\dfrac{z-w}{z(1-\bar{z}w)}\right|=
\left|\dfrac{z-w}{z-z\bar{z}w}\right|=
\left|\dfrac{z-w}{z-w}\right|=1
$$
If $|w|=1$, consider that $|1-\bar{z}w|=|1-\bar{w}z|$ because they are conjugate.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/779962",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Prove that the union of two disjoint countable sets is countable This is a question from my proofs course review list that I have had trouble understanding.
I understand the concept of disjoint sets. I'm not sure what they mean by countable. How would one prove a set is countable and furthermore, that the union of tw... | (In the following, countable means what is often referred to as countably infinite, i.e. countable and not finite)
A set $A$ is called countable if there exists a bijection $f$ between $A$ and the set of natural numbers $\mathbb{N}$. In other words, $A$ is countable if there is some mapping $f \,:\, A \to \mathbb{N}$ s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/780043",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Calculating the residues of $\frac{\zeta^{\prime}{(s) x^{s}}}{\zeta(s)\cdot s}$ Calculating the poles of $\frac{\zeta^{\prime}{(s) x^{s}}}{\zeta(s)\cdot s}$, where x is a fixed real number.
I am trying to calculate the poles of this function at the trivial zeros of $\zeta$, namely the even negative integers.
To do so... | For a fixed $x\in \mathbb{R}\setminus \{0\}$, $s\mapsto x^s$ is an entire function (you can choose different branches of that function by choosing different logarithms of $x$, but they all are entire), so that factor contributes no poles. In simple poles of the function, it modifies the residue just by multiplication w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/780128",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Cartier divisors and global sections I have a brief question - I seem to have a vague recollection that if we have a Cartier divisor $D$ on a scheme $X$ , then we can determine whether $D$ is effective by saying whether $\mathcal{O}_X(D)$ has a global section or not. I have tried to prove this fact, but can't seem to d... | Effective cartier divisors are just closed subschemes which locally are cut out by the vanishing of a non-zero divisor.
I don't know what you mean by $\mathcal{O}_X(D)$, but if you mean $\mathcal{O}_D(D)$, then certainly it need not have a nonconstant global section. Take for example a family of elliptic curves over $\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/780207",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
How to show these inequalities? To show that $\forall n \in \mathbb{N}$
$$n \log{n}-n+1 \leq \log{(n!)} \leq (n+1) \log{n}-n+1$$
do I have to use induction? Once at the one inequality and then at the other?
Or is there an other way to show this?
| $$\log n!=\log 1+\log 2+\cdots +\log n$$ so $$\log n!\le \int_1^{n+1}\ln x\,\mathrm{d}x=(n+1)\ln(n+1)-(n+1)+1=(n+1)\ln(n+1)-n$$ and perhaps you can toy with that to get the right side.
For the other direction, $$\log n!=\log 2+\cdots +\log n\ge \int_1^n \ln x\,\mathrm{d}x=n\ln n-n+1$$
In general for an increasing funct... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/780305",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What is the reason for this geometric answer?
$$\frac{AB}{CD} = \frac{2}{2.6} = 0.77\ldots$$
$$\frac{AC}{AD} = \frac{2}{2.6} = 0.77\ldots$$
$$\frac{BC}{AC} = \frac{2}{2.6} = 0.77\ldots$$
Therefore $\triangle ABC$ and $\triangle ACD$ are similar.
I know from the answer sheet that $y$ is $47^\circ$ and $x$ is $109^\circ... | Continuing,
since the triangles are similar,
$x = 109$
and
$y
= CAD
=180-(109+47)
=180-156
=24
$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/780367",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Problem understanding the Axiom of Foundation I am just beginning to learn the ZF axioms of set theory, and I am having trouble understanding the Axiom of Foundation. What exactly does it mean that "every non-empty set $x$ contains a member $y$ such that $x$ and $y$ are disjoint sets." In particular, how can $y$ be an ... | The axiom isn't saying that $x\cap \{y\}=\emptyset$. It's saying $x\cap y=\emptyset$. Keep in mind that in ZFC everything is a set, including the elements of other sets.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/780429",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
How to prove the inequalities $\int_{0}^{1}\sin{(x^n)}dx\ge\int_{0}^{1}(\sin x)^ndx\ge 0$
Show that:
$$\int_{0}^{1}\sin{(x^n)}dx\ge\int_{0}^{1}(\sin x)^ndx\ge 0$$
My idea:maybe $\sin{(x^n)}\ge (\sin{x})^n？$
| Let $n\geqslant1$ and $u(x)=\sin(x^n)-(\sin x)^n$ then $$u'(x)=n\cdot\left(x^{n-1}\cos(x^n)-(\sin x)^{n-1}\cos x\right).$$ For every $x$ in $(0,1)$, $x^n\leqslant x$ hence $\cos(x^n)\geqslant\cos x$, and $x\geqslant\sin x$ hence $x^{n-1}\geqslant(\sin x)^{n-1}$. Thus, $u'(x)\geqslant0$ for every $x$ in $(0,1)$. Since $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/780655",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
} |
Learning Advanced Mathematics I'm a 12th grade student and I've recently developed a passion for mathematics . Currently my knowledge in this particular area is comprised by : single-variable calculus , trigonometry , geometry , basic notions of linear algebra and set theory .
I'm particularly interested in calculus a... | As DonAntonio commented a good idea is to review single variable calculus emphasizing the theorems and their proofs. A good book to go beyond just the praxis of solving integrals and into the demonstrations is Tom Apostol's "Calculus and Linear Algebra" volumes 1 and 2. The first is primarily concerned with single vari... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/780782",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to find PV $\int_0^\infty \frac{\log \cos^2 \alpha x}{\beta^2-x^2} \, \mathrm dx=\alpha \pi$ $$
I:=PV\int_0^\infty \frac{\log\left(\cos^2\left(\alpha x\right)\right)}{\beta^2-x^2} \, \mathrm dx=\alpha \pi,\qquad \alpha>0,\ \beta\in \mathbb{R}.$$
I am trying to solve this integral, I edited and added in Principle v... | Consider the function $$ f(z) = \frac{\log(1+e^{2i \alpha z})}{z^{2}-\beta^{2}} \ , \ (\alpha,\beta >0)$$
which is well-defined on the complex plane if we omit the real axis and restrict $z$ to the upper half-plane while defining $\log (1+e^{2iaz})$ to be $\log(2)$ just above the origin.
Notice that $$\text{Re} \big( ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/781017",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "29",
"answer_count": 3,
"answer_id": 1
} |
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