Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
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Solving for positive reals: $abcd=1$, $a+b+c+d=28$, $ac+bc+cd+da+ac+bd=82/3$ $$a,b,c,d \in \mathbb{R}^{+}$$
$$ a+b+c+d=28$$
$$ ab+bc+cd+da+ac+bd=\frac{82}{3} $$
$$ abcd = 1 $$
One can also look for the roots of polynomial
$$\begin{align}
f(x) &= (x-a)(x-b)(x-c)(x-d) \\[4pt]
&= x^4 - 28x^3 + \frac{82}{3}x^2 - (abc+abd+a... | Assume $d = \max{a,b,c,d}$. Looking at the inequality:
$$(a+b+c)^2\geq 3(ab+bc+ca)$$
beginning edit by Will:
from Michael,
$$ 82 = 3 (bc+ca+ab) + 3d(a+b+c), $$
from displayed inequality
$$ 82 \leq (a+b+c)^2 + 3d(a+b+c) $$
$$ 82 \leq (28-d)^2 + 3 d (28-d) $$
$$ 82 \leq 784 - 56d + d^2 + 84d - 3 d^2 $$
$$ 0 \leq 702 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3764736",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 2
} |
Newton's evaluation of $1 + \frac{1}{3} - \frac{1}{5} - \frac{1}{7} + \frac{1}{9} + \frac{1}{11} - \cdots$ How might have Newton evaluated the following series?
$$\sqrt{2} \, \frac{\pi}{4} = 1 + \frac{1}{3} - \frac{1}{5} - \frac{1}{7} + \frac{1}{9} + \frac{1}{11} - \cdots$$
The method of the this thread applies by sett... | Although the question appears to be about how Newton historically did it, I'll convert a popular comment to an answer showing how techniques from his era, similar to those that handle the Gregory series, evaluate the series above:
$$\begin{align}\sum_{n\ge0}\left(\frac{1}{8n+1}+\frac{1}{8n+3}-\frac{1}{8n+5}-\frac{1}{8n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3764826",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to solve this ODE: $y'(x) e^x = y^2(x)$? I am trying to solve the differential equation
$$y'(x) e^x = y^2(x) \quad (DE) $$
This is a Bernoulli form DE i.e $y'(x) + a(x)y(x) = b(x)y^r(x)$, where $r = 2, a(x) = 0, b(x) = \frac1e $
*
*Let $u(x) = y^{1-r} = y^{-1} \iff u'(x) = -y^{-2}(x) y'(x)$
*Then for $y \neq 0$... | Your equation is equivalent to
$$\frac{\dot{y}}{y^2}=e^{-x}$$
as long as $y(t)\neq0$. (Notice that $y(t)\equiv0$ is a solution to your problem)
Integrating over some intervals,say $[x_0,x]$ leads to
$$
\int^x_{x_0}\frac{y'(t)}{y^2(t)}\,dt=\int^x_{x_0}e^{-t}\,dt=-e^{-t}|^x_{x_0}=e^{-x_0}-e^{-x}
$$
The integral on the le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3765058",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Calculation of $\left(\frac{1}{\cos^2x}\right)^{\frac{1}{2}}$ Shouldn't $\left(\frac{1}{\cos^2x}\right)^{\frac{1}{2}} = |\sec(x)|$?
Why does Symbolab as well as my professor (page one, also below) claim that $\left(\frac{1}{\cos^2x}\right)^{\frac{1}{2}} = \sec(x)$, which can be negative? Also, the length of a vector ca... | You are correct that $\sqrt{x^2} \neq x$ for every $x\in\Bbb R$. Otherwise, we would have absurdity like $1 = \sqrt{1}=\sqrt{(-1)^2} =-1$. For $x\in\Bbb R$ we indeed have $\sqrt{x^2}=|x|$. However, we have $\sqrt{x^2}=x$ for all $x\geq 0$.
Hence, as you professor seems to assume that $t\in [-\pi/2,\pi/2]$, his claim th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3765202",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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Does local cohomology commutes with direct sums? Let $A$ be a commutative noetherian ring, $I\subseteq A$ an ideal, $M_\alpha$ be $A$-modules, $\forall\alpha\in J$. It is easily seen that the $I$-torsion commutes with direct sums:
$$\Gamma_I(\bigoplus_{\alpha\in J}M_\alpha)=\bigoplus_{\alpha\in J}\Gamma_I(M_\alpha).$$
... | Convince yourself first that $H_I^n(M) = \varinjlim_k \operatorname{Ext}_R^n(R / I^k, M);$ then, use the fact that Ext commutes with finite direct sums in the second component, i.e., $$\operatorname{Ext}_R^n(R / I^k, \oplus_{i = 1}^m M_i) \cong \oplus_{i = 1}^m \operatorname{Ext}_R^n(R / I^k, M_i).$$
For the first fact... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Show that if the matrix of $T$ with respect to all bases of $V$ is same, then $T = \alpha I$ where $T$ is a linear operator on $V$ Now the only hint I could derive from the question that we might have to use eigenvalues as $Tx = \lambda x$ for $T = \lambda I$. I think eigenvalues are invariant with basis.
Any help woul... | It seems that this is related to the isotropic matrices, i.e. matrices proportional to the identity. When you use a change of basis you transform $T$ into something else (in general components of the transformed matrix are different, but this does not occur in here). Recall the transformation of $T$ into $Q$: $Q=V^{-1... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to prove that $\int_0^1 f(x)\,dx = f(0) + \frac{1}{2}f'(c)$ for some $ c \in [0,1]$?
*
*Given that ${f}$ is differentiable on the interval
$[0,1]$ I need to prove that $\int_0^1 f(x)dx = f(0) + \frac{1}{2} f'(c)$ for some $ c \in [0,1]$.
*I'm aware of integral mean value theorem, which gives us the following:
E... | Let $\;a=\int_0^1 f(x)\,dx-f(0)$.
It is sufficient to prove that there exists $c\in\left]0,1\right[$ such that $a=\frac{1}{2}f’(c)$.
We define the following function:
$$\phi(t)=\int_0^t f(x)\,dx+(1-t)f(t)+a(1-t)^2:[0,1]\to\mathbb{R}$$
$\phi(t)$ is differentiable on $[0,1]$ and
$$\phi'(t)=f(t)-f(t)+(1-t)f’(t)-2a(1-t)=(1... | {
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Does ${f(x)=\ln(e^{x^2})}$ reduce to ${x^2\ln(e)}$ or ${2x\ln(e)}$? I'm confused with the expression ${f(x) = \ln(e^{x^2})}$.I know the rule ${\log_a(x^p) = p\log_a(x)}$. So does the given expression reduce to ${x^2\ln(e)}$ or ${2x\ln(e)}$?
| Here you have $\ln(e^p)$ with $p = x^2$, so the correct reduction is $(x^2)\ln(e) = x^2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3765701",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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Why does every oscillating sequence diverge? Is this statement true?
$$\bf\text{Every oscillating sequence diverges.}$$
My thoughts: $\bf{False}$. $s_n = (-1)^n$ does not converge. But it's bounded, therefore, not divergent either. Divergent means diverging to $-\infty$ or $+ \infty$, yes?
Solution key: $\bf{True}$. If... | In the usual definitions that I have picked up from various instructors and textbooks, "this sequence diverges" just means "this sequence doesn't converge". You could self-consistently define things the other way, but this is unusual in my experience.
By contrast, "this sequence oscillates" is usually not rigorously de... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3765818",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Solution to autonomous differential equation with locally lipscitz function As I was learning about the following theorem and its proof from the book Nonlinear Systems by H. K. Khalil, I encountered a difficulty in grasping some parts of the proof.
Theorem: Consider the scalar autonomous differential equation
\begin{eq... | That's not what it's saying. It's saying $y(t) \to 0$ can't happen in finite time, i.e. there can't be a solution $Y(t)$ of the differential equation with $Y(t_0) = y_0$ and $Y(t_1) = 0$ for some $t_1 > t_0$.
Suppose that did happen.
Note that $y(t) = 0$ is also a solution of the differential equation, because part of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3765911",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Do Approximate Eigenvalues Imply Approximate Eigenvectors? My apologies in advance if this has already been asked somewhere.
Suppose I have two real symmetric matrices $A$ and $B$ in $\mathbb{R}^{d \times d}$ for which $\lVert A - B \rVert_{op} \le \varepsilon$. Further, call the eigenvalue-eigenvector pairs for $A$ an... | Having $\|A - B\|$ small is not enough, in itself, to make
$u_i$ and $v_i$ close. Consider any real symmetric matrices $A_0$ and $B_0$ (with distinct eigenvalues to avoid any problems of degeneracy), and take $A = t A_0$ and $B = t B_0$. Thus
$\|A - B\| = |t| \|A_0 - B_0\|$ can be made arbitrarily small by taking $t$... | {
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Simple proof of: if $ax\equiv ay \pmod{m}$, and $\gcd(a,m)=1$, then $x\equiv y$ I'm working on a proof of: "if $ax\equiv ay \pmod{m}$, and $\gcd(a,m)=1$, then $x\equiv y\pmod{m}$". Here's what I have so far:
Suppose $ax\equiv ay\pmod{m}$, and $\gcd(a,m)=1$
By definition, $ax = ay + mp$ for some $p\in\mathbb{Z}$
By defi... | The proof you gave may have a flaw: if $1=gcd(a,m)=ax=ay$, then $|a|=|x|=|y|=1$, which is not the case. By Bezout's Identity, from $ ax=ay+mp$ and $ay=ax+mr$, we can only imply $ax$ and $ay$ are multiplier of $gcd(a,m)$
The proposition you stated is a special case of a general proposition:
if $ax\equiv ay (mod$ $m)$, ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Proving that $\mathbb{Z}[i]/\langle 2+3i\rangle $ is a finite field Prove that $\mathbb{Z}[i]/\langle 2+3i\rangle $ is a finite field.
Hi. I can't try a few steps in the next solution
$$\mathbb{Z}[i]/\langle 2+3i\rangle \simeq \mathbb{Z}[x]/\langle 1+x^2,2+3x\rangle$$ and $9(1+x^2)+(2-3x)(2+3x)=13$ then $13\in \langle ... | Question 1: We have $\Bbb Z[i]\simeq \Bbb Z[x]/\langle x^2+1\rangle$. And the third isomorphism theorem says that when we are dividing out by first $x^2+1$, and then $2+3x$, we are allowed to divide out but both of them simultaneously.
Question 2: Again justified by the third isomorphism theorem, dividing out by $13$ b... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that $V=Z(x;T)\oplus Z(y;T)$ and the $T$-annihilators $\mu_{T,x},\,\mu_{T,y}$ do not share any common divisors implies that $V$ is cyclic Provided that $V=Z(x;T)\oplus Z(y;T)$ where $Z(v;T)$ denotes the cyclic subspace and the corresponding $T$-annihilators $\mu_{T,x},\,\mu_{T,y}$ do not share any common divisors,... | Hint: It suffices to show that $Z(x+y;T)$ contains both $x$ and $y$. To that end, note that the restrictions $\mu_{T,x}(T) \mid_{Z(y;T)},\mu_{T,y}(T) \mid_{Z(x;T)}$ are invertible.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3766428",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why is the spectrum of a shift operator the closed unit disk? Consider the following text from Murphy's: "$C^*$-algebras and operator theory":
In example 2.3.2, why is $\sigma(u) = \Bbb{D}$ (= the closed unit disk)?
I can see that $\sigma(u) \subseteq \Bbb{D}$ and $\sigma(u^*) = \Bbb{D}.$
Thanks in advance!
| $$ \lambda \in \sigma(u) \iff \overline{\lambda} \in \sigma( u^*).$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3766576",
"timestamp": "2023-03-29T00:00:00",
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How to find the set of lines of fixed length connecting two circles? Suppose I have two circles as shown below.
I want to find the set of line segments of size $\mathbf{w}$ connecting the two circles.
What's a nice way to do this?
I thought of a simple, ugly way. I could write out the equations
\begin{align}
(u_x - a... | Not a complete answer
It's not going to be pretty no matter how you go about it, but I'd recommend a couple of steps.
*
*Rotate and translate until $(a, b) = (0,0)$ and $(c, d) = (c', 0)$. Then scale everything so that $\|u\| = 1$ (although maybe this step isn't needed).
*Observe that there may be no solutions. If ... | {
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"timestamp": "2023-03-29T00:00:00",
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How can I prove $A − A(A + B)^{−1}A = B − B(A + B)^{−1}B$ for matrices $A$ and $B$? The matrix cookbook (page 16) offers this amazing result:
$$A − A(A + B)^{−1}A = B − B(A + B)^{−1}B$$
This seems to be too unbelievable to be true and I can't seem to prove it. Can anyone verify this equation/offer proof?
| \begin{align}
A - A(A+B)^{-1}A & = A(A+B)^{-1}(A+B) - A(A+B)^{-1}A \\ &= A(A+B)^{-1}(A+B - A)\\ &= A(A+B)^{-1}B \\ &= (A+B - B)(A+B)^{-1}B \\ &= (A+B)(A+B)^{-1}B - B(A+B)^{-1}B \\ &= B - B(A+B)^{-1}B
\end{align}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3766930",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Show that the residue is $c_{-1}=-\frac{q''(z_0)}{(q'(z_0))^3}$. Suppose that
$f(z) = \frac{1}{(q(z))^2}$ where the function $q$ is analytic at $z_0$, $q(z_0) = 0$, and $q'(z_0)\neq 0$. Show the residue is $c_{-1}=-\frac{q''(z_0)}{(q'(z_0))^3}$.
First I would like to acknowledge that there is an old question on stack e... | Why can we do that?
It can be done with a simple substitution: set $t=z-z_0$. Then $z=z_0\iff t=0$, and the function $f(z)$ becomes the function $g(t)=\dfrac 1{\bigl(q(z_0+t)\bigr)^2}$
Is this a Laurent series? What does the $O$ mean?
It is not a Laurent series, since it is not a series. It is just the Taylor's expan... | {
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How should one understand the "indefinite integral" notation $\int f(x)\;dx$ in calculus? In calculus, it is said that
$$
\int f(x)\; dx=F(x)\quad\text{means}\quad F'(x)=f(x)\tag{1}
$$
where $F$ is a differentiable function on some open integral $I$. But the mean value theorem implies that any differentiable function ... | I think the problem is not only for antiderivative, but more generally it has to cope with the abuse of notation for multivalued functions.
Take for instance the complex logarithm $\ln(z)=\overbrace{\ln(r)+i\theta}^{\operatorname{Ln}(z)}+i2k\pi$.
You have to understand $$\ln(z_1z_2)\color{red}=\ln(z_1)+\ln(z_2)$$
As $$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3767159",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Uniform motion question A biker and a skater set out at 4:30 PM from the same point, headed in the same direction. The biker is travelling at a rate of 15 km/hr faster than twice the speed of the skater. In 1.5 hours, the biker is 35 km ahead of the skater. Find the rate of the skater.
My approach:
Biker speed = $S_1$,... | The equation of motion for the biker is $x_1(t)=(2v+15) t$. The equation of motion of the skater is $x_2(t)=vt.$ We know that $x_1(1.5)-x_2(1.5)=35.$ Thus
$$(2v+15)\cdot 1.5-1.5v=35$$
$$1.5(2v+15-v)=35$$
$$v+15=\frac{70}{3}$$
$$v=\frac{25}{3}~\text{km}/\text{hr}.$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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What does distance of a point from line being negative signify? When we take distance from the line, we take
$$ d = \frac{ Ax_o + By_o + C}{ \sqrt{A^2 +B^2}}$$
usually with a modulus on top, now my question is if I evaluate this distance as negative what does it mean? Can I decide on which half-plane a point using this... | Expanding on the answer of @Andrew Chin, the sign says in which of the two half-planes (in which the line divides the whole plane) lays the point. If it is from the same side of the vector $(A,B)$ then the result is positive, otherwise negative. Sometime called oriented distance.
For example, for the line of equation $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3767405",
"timestamp": "2023-03-29T00:00:00",
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$(A^c\times B)\cup(A\times B^c)\cup(A^c \times B^c)=(A^c\times Y)\cup(X\times B^c)$ Let $(X,T)$ and $(Y,T')$ be two topological spaces and $A\subset X ,B\subset Y$.Show that $$(A^c\times B)\cup(A\times B^c)\cup(A^c \times B^c)=(A^c\times Y)\cup(X\times B^c)$$
solution $$1...........(A^c\times B)\cup(A\times B^c)\cup(A^... | Already in the first step you’ve written something that suggests a misconception on your part. It’s true that $(A\times B^c)\cup(A^c\times B^c)=(A\cup A^c)\times(B^c\cup B^c)$, but only because $(A\times B^c)\cup(A^c\times B^c)=(A\cup A^c)\times B^c$; it is not in general true that
$$(W\times X)\cup(Y\times Z)=(W\cup Y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3767920",
"timestamp": "2023-03-29T00:00:00",
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An illusionist and their assistant are about to perform the following magic trick
Let $k$ be a positive integer. A spectator is given $n=k!+k−1$ balls numbered $1,2,\dotsc,n$. Unseen by the illusionist, the spectator arranges the balls into a sequence as they see fit. The assistant studies the sequence, chooses some b... | NOTE: I found counterexamples to the explicit $f$ I initially posted. I removed it but am leaving the rest of the answer up as a partial solution.
Notation and Remarks
Let $S_n$ denote the set of permutations of length $n$ and let $C_{n,k}$ be the set of covered permutations. For example, the permutation $12345678 \in... | {
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"timestamp": "2023-03-29T00:00:00",
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Line Integral gives no work done? For the following question,
$$
\mathbf{F}=\langle-y, x\rangle
$$
For this field:
Compute the line integral along the path that goes from (0,0) to (1,1) by first going along the $x$ -axis to (1,0) and then going up one unit to (1,1) .
I got an answer of $0$, by doing:
But the answer k... | Here's how physicists often do it:
$$\int_C \mathbf{F}\cdot d\mathbf{r} = \int_C (F_x \hat{i} + F_y \hat{y})\cdot (\hat{i}\,dx + \hat{j}\,dy) = \int_C F_x\,dx + F_y\,dy$$
On the first part $x$ goes $0 \to 1$ and $y=0$ is constant so $dy = 0$ and hence
$$\int_{C_1} F_x\,dx + F_y\,dy = \int_{x=0}^{x=1} -y\,dx = 0.$$
Simi... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Can $a \bmod 3$ be represented arithmetically without the mod or other integer-related functions? I've noticed that $a \bmod b$ (with 'mod' in the operational sense) can also be represented using various tricks of formulation. For instance, that
$$a \bmod b = a - b\left\lfloor\frac{a}{b}\right\rfloor .$$
There are seve... | Note that $\sin(2\pi n/3)=0, \dfrac{\sqrt3}2, $ or $-\dfrac{\sqrt3}2$, according as $n\equiv0, 1, $ or $2\bmod3$, respectively.
Furthermore, $f(x)=\dfrac{x\left(x-\dfrac{\sqrt3}2\right)2}{-\dfrac{\sqrt3}2\left(\dfrac{\sqrt3}2-\dfrac{\sqrt3}2\right)}+\dfrac{x\left(x+\dfrac{\sqrt3}2\right)1}{\dfrac{\sqrt3}2\left(\dfrac{\... | {
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Evaluating $\iint dx\,dy$ over the region bounded by $y^2=x$ and $x^2+y^2=2x$ in the first quadrant
Identify the region bounded by the curves $y^2=x$ and $x^2+y^2=2x$, that lies in the first quadrant and evaluate $\iint dx\,dy$ over this region.
In my book the solution is like:
$$\begin{align}\\
\iint dx\,dy &=\int_... | Clearly, your area cannot be negative, so your result is immediately incorrect.
The system $$x = y^2 \\ x^2 + y^2 = 2x$$ is readily solved by substitution. We have $$\begin{align}
0 &= x^2 + y^2 - 2x \\
&= x^2 + x - 2x \\
&= x^2 - x \\
&= x(x-1).
\end{align}$$ Hence $x \in \{0, 1\}$ and the full solution set is $$(x,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3768368",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Evaluate $\int \frac{2-x^3}{(1+x^3)^{3/2}} dx$
Evaluate:
$$\int \frac{2-x^3}{(1+x^3)^{3/2}} dx$$
I could find the integral by setting it equal to $$\frac{ax+b}{(1+x^3)^{1/2}}$$
and differentiating both sides w.r.t.$x$ as
$$\frac{2-x^3}{(1+x^3)^{3/2}}=\frac{a(1+x^3)^{3/2}-(1/2)(ax+b)3x^2(1+x^3)^{-1/2}}{(1+x^3)}$$$$=\f... | @ClaudeLeibovici has a point, because what you did is a well-worn technique, that of using an Ansatz. The basic idea is to make an educated guess as to the form of the solution, then make it more specific with your calculations, as you did. So it's worth understanding what makes a specific Ansatz a sensible starting po... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3768479",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Show that the solution of the equation $x^5-2x^3-3=0$ are all less than 2 (using proof by contradiction).
The question is "Show that the solution of the equation $x^5-2x^3-3=0$ are all less than 2."
I have attempted to answer this question using proof by contradiction and I think my answer is either wrong or not a we... | Let $x$ be a root and $x>2$.
Thus, $$x^5-2x^3-3=x^5-2x^4+2x^4-4x^3+2x^3-3>0,$$ which is a contradiction.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3768647",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If $A$ is a rank $1$ matrix, then $A^2= \operatorname{Tr}A \cdot A$ I am posting this question because I want to know if my proof is correct (I know that the result holds for $\mathbb{K}=\mathbb{C}$ and I see no reason why it wouldn't work for an arbitrary field, but I just want to be sure).
Claim : Let $A \in \mathcal... | It's technically correct, but I think it's clearer to write $A$ as $bc^T$ with $b,c\in\mathbb K^n$. The identity can then be proved easily as
$$
A^2=(bc^T)(bc^T)=b(c^Tb)c^T=(c^Tb)bc^T=(\operatorname{Tr}A)A.
$$
If you decompose $A$ into a product of two square matrices $B$ and $C$, one may not immediately see why $B(CB)... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Calculus midterm exam help. I received my grade for my calculus midterm and according to my professor I answered the two following questions incorrectly but he refuses to tell me why. If anyone could explain to me where I went wrong I would appreciate it greatly.
According to my professor, the proof used in the fir... | For question 1
I imagine that your professor was expecting you to provide a cartesian equation of the plane in the form $ax + by +cz = 0$. So you should expand what you wrote with $ \vec r = x \vec i + y \vec j +z \vec k$.
For question 2
Your initial speed is incorrect and equal to $6 \vec j$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3768865",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Does a symmetric diagonally dominant real matrix $A$ with nonnegative diagonal entries satisfy $(x^{2p-1})^T A x \geq 0$? In https://mathworld.wolfram.com/DiagonallyDominantMatrix.html, I find that
A symmetric diagonally dominant real matrix with nonnegative diagonal entries is positive semidefinite.
If $A \in \mathb... | This is true. Denote the standard basis of $\mathbb R^N$ by $e_1,e_2,\ldots,e_N\}$. If $A$ has a nonzero off-diagonal entry $a_{ij}$, let
$$
B=|a_{ij}|\left(e_i+\operatorname{sign}(a_{ij})e_j\right)\left(e_i+\operatorname{sign}(a_{ij})e_j\right)^T.
$$
Then both $B$ and $A-B$ are diagonally dominant and their diagonals ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Prove $\sum_{n=1}^\infty a_n b_n $ is convergent if $\sum_{n=1}^\infty (b_n -b_{n+1})$ is absolutely convergent , $\sum a_n $ convergent Prove that $\sum_{n=1}^\infty a_n b_n $ is convergent if $\sum_{n=1}^\infty a_n$ is convergent and $\sum_{n=1}^\infty (b_n -b_{n+1})$ is absolutely convergent series.
Since, $\sum_{n=... | The fact that $\sum_n(b_n-b_{n+1})$ converges means $b_1-b_{n+1}=\sum_{k=1}^n(b_k-b_{k+1})$ converges, hence $b_n$ converges. Similarly, $A_n:=\sum_{k=1}^na_k$ converges. Hence both sequences $A_n$ and $b_n$ are bounded by, say, $A$ and $B$ respectively, and both are Cauchy sequences.
$$\sum_{n=p}^qa_nb_n=\sum_{n=p}^qA... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Derive $\mathbf b$ from $\mathbf a = \mathbf b× \mathbf c$ I have an equation:
$$\mathbf a = \mathbf b × \mathbf c,$$
where $\mathbf a$ $\mathbf b$ and $\mathbf c$ are 3-vectors.
How could I derive $b$ from the equation and express it in terms of $\mathbf a$ and $\mathbf c$?
| Note that
*
*if $\mathbf{c}\cdot\mathbf{a}\neq 0$ then there are no solutions.
*if $\mathbf{c}=\mathbf{0}$, then there are no solutions unless $\mathbf{a}=\mathbf0$, in which case every $\mathbf{b}$ works.
*if $\mathbf{c}\neq\mathbf{0}$, then $\mathbf{p}\times\mathbf{c}=\mathbf{0}$ if and only if $\mathbf{p}$ is a ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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F is a set of 101 functions from[10] to itself. Prove that $\exists f,g \in F$ and $\exists i,j \in [10]$ such that $f(i) = g(i)$ and $f(j) = g(j)$.
$F$ is a set of $101$ functions from $\{1,2,\ldots,10\}$ to itself.
Prove that $\exists f,g\in F$ and $\exists i,j\in\{1,2,\ldots,10\}=[10]$
so $f(i)=g(i)$ and $f(j)=g(j)... | When $m$ and $n$ are positive integers, how many distinct functions are there from $\{1,2,...,n\}$ to $\{1,2,...,m\}$?
To define a function $f:\{1,...,n\}\to\{1,...,m\}$ means that for each element $i\in\{1,...,n\}$ we have specified its image $f(i)\in\{1,...,m\}$. Since the codomain has $m$ many elements, there are $m... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Discrete null recurrent Markov chain implies that $\lim_{n\to \infty} P(X_n=j) = 0$ In the book “Markov Chains” (by Norris), in Theorem 1.8.5 the author proves that if the chain is aperiodic irreducible and null recurrent, then $\lim_{n \to \infty} P(X_n = j) = 0$. I’m having some trouble understanding one of the steps... | The first inequality $1\geq \sum_{k=n-K+1}^n P(X_k =j) P(T_j >n-k\mid X_0=j)$ follows from the fact that the right hand is actually a probability. It's the probability of hitting $j$ at some time $k$ for the first time and then not returning to $j$ before a time $n-k$ has passed. The sum is over disjoint sets since we ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Evaluate the following integral $ \int_1^{\infty} \frac{\lbrace x\rbrace-\frac{1}2}{x} dx$ $$\int_1^{\infty} \frac{\lbrace x\rbrace-\frac{1}2}{x} dx$$
Here $\lbrace\cdot\rbrace$ denotes the fractional part.
I found this challenging integral, and I'm curious about the solution, so I decided to do some efforts to solve i... | This function is not integrable in the Lebesgue sense, so you can only evaluate the Cauchy principal value.
That is, what you want to evaluate is the limit $$\lim_{M \rightarrow +\infty} \int_1^M \frac{\{x\}-\frac12}xdx.$$
It is easy to see that it suffices to take the limit for integer values of $M$. We first compute,... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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How many distinct permutations of the string "NADAMADRID" have the word DAM appearing in them? How many distinct permutations of the string "NADAMADRID" have the word DAM appearing in them?
Normally, under the Mississippi Rule, you would take the total number of characters factorial, then divide by the product of all ... | If we discard the word $DAM$ temporarily, we have $7$ letters to arrange, out of which two letters of two types are similar. The arrangement of these can be done in $\frac{7!}{(2!)^2}$ ways. Once this is done, we have $8$ spaces for the word $DAM$ to be inserted into , so in total you get $$\frac{7!}{(2!)^2} \times 8 =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3770168",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Proofs by Induction: are my two proofs correct? I have been trying to understand how proof by mathematical induction works, and I am struggling a bit. But, I think I am understanding it and I just want to verify that what I am doing is correct (and if not, why?)
I have attached a screenshot (as a link) of my problem (b... | In a proof by mathematical induction, we wish to establish that some property $P(n)$ holds for each positive integer $n$ (or for each integer greater than some fixed integer $n_0$). We must first establish that the base case holds. Once we establish that it holds, we may assume the property holds for some positive in... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Prove: $(n-1)! \equiv -1 \pmod{n}$ I'm trying to self-learn some number theory. Currently, I'm trying to prove the following:
$$(n-1)! \equiv -1 \pmod{n}$$
However, I'm a bit stumped on this one. Honestly, I'm not sure where's a good place to start.
| This is false for general $n$. For example, $5!=120\equiv 0\pmod 6$, not $-1$. In fact, $(n-1)!\equiv 0\pmod{n}$ for all composite $n\geq 6$.
| {
"language": "en",
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Corollary 5.39, Lee - Introduction to Smooth Manifolds I am struggling with understanding how to prove Corollary 5.39 in Lee - Introduction to Smooth Manifolds.
I found an answer on this post: A characterization of tangent space to level set of a smooth submersion, but I do not understand the computation done there. W... | We know that $d\Phi_p^i(v)$ is a vector in $T_{\Phi^i(p)}\mathbb{R} = \text{span }\Big(\frac{d}{dt}\big|_{\Phi^i(p)}\Big)$. So $d\Phi^i_p(v) = a \frac{d}{dt}\big|_{\Phi^i(p)}$ for some real number $a$. Since we often identify one dimensional vector $\lambda \, \frac{d}{dt}\big|_{t}$ with its component $\lambda$ itself,... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Painting of a block in P&C Consider a grid as shown in figure having 8 blocks of dimension 1 × 1. Each block is to be painted with
either blue or red colour. In how many ways this can be done if at least one block of dimension 2 × 2 is
to be painted red completely ?
I am not able to approach this problem. Can this pro... | Define sets $A,B,C$ as follows . . .
*
*Let $A$ be the set of configurations for which the upper left $2{\times}2$ submatrix is painted red.$\\[4pt]$
*Let $B$ be the set of configurations for which the upper right $2{\times}2$ submatrix is painted red.$\\[4pt]$
*Let $C$ be the set of configurations for which the l... | {
"language": "en",
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The quadratic equations $x^2+mx-n=0$ and $x^2-mx+n=0$ have integer roots. Prove that $n$ is divisible by $6$.
QUESTION: Suppose that $m$ and $n$ are integers, such that both the quadratic equations $$x^2+mx-n=0$$ and $$x^2-mx+n=0$$ have integer roots. Prove that $n$ is divisible by $6$.
MY APPROACH:
$\because$ the r... | Now as you got $m^2+4n=p^2$ and $m^2-4n=q^2$, we solve further by taking cases.
Case 1: $m$ is even
Therefore let $m=2k$ for some positive integer $k$ and by judging the equation we can see that $p$ and $q$ are even too. Let $p=2a$ and $q=2b$ for some positive integers $a$ and $b$. Substituting the values of $m,p$ and ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3771027",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Let $A_1 \cap A_2 \cap \dots \cap A_n \neq \varnothing $. Then $A_1 \cup A_2 \cup \dots \cup A_n \neq \varnothing$. Let $A_1,A_2,\dots, A_n$ be sets such that $A_1 \cap A_2 \cap \dots \cap A_n \neq \varnothing $ holds for all $n$. Then $A_1 \cup A_2 \cup \dots \cup A_n \neq \varnothing$.
Is the following proof correct?... | Writing formal details:
$$A_1 \cap A_2 \cap \dots \cap A_n \neq \varnothing \Rightarrow\\\Rightarrow \exists x \in A_1 \cap A_2 \cap \dots \cap A_n \Rightarrow\\\Rightarrow \exists i, 1 \leqslant i \leqslant n, x \in A_i \Rightarrow\\ \Rightarrow\ x \in A_1 \cup A_2 \cup \dots \cup A_n \Rightarrow\\ \Rightarrow A_1 \cu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3771155",
"timestamp": "2023-03-29T00:00:00",
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Equal number of $n$th roots of unity within character values $f_1(a), \dots, f_m(a)$. (Apostol exercise 6.12 Intro to ANT) This problem is taken from Exercise 6.12 from Apostol's "Introduction to Analytic Number Theory".
Verbatim, the problem states
Let $f_1, \dots, f_m$ be the characters of a finite group $G$ of orde... | Here is a proof, using only facts from Chapter 6 of Apostol, and using its notation. Let $a\in G$ have order $n$.
The characters form a group under multiplication. Therefore, if $f_1,\dots,f_r$ are the characters, then multiplying by $f_i$ simply permutes the characters. In particular, for any fixed $a\in G$, it permut... | {
"language": "en",
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Is a function increasing if the derivative is positive except at one point of an interval? Let $f: \mathbb{R} \to \mathbb{R}$ be differentiable on $(a,b)$. Suppose $f' > 0$ on $(a,b)$ except at a point $c \in (a,b)$ (that is, $f'(c) \leq 0$).
*
*Is $f$ increasing on $(a,b)$?
*Must $f'(c)$ be zero, or can it be negat... | You can use the fact that the derivative has intermediate value property to rule out $f'(c)<0$.(this will contradict that $c$ is only point where derivative is not positive)
Now pick a $x<c$ then by MVT there is $\eta\in(x,c)$ such that $f(c)-f(x)=f'(\eta)(c-x)>0$ [this is actually independent of derivative being defin... | {
"language": "en",
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Why isn't projection of $\mathbf{v}$ onto $\mathbf{u}$ defined so that it's perpendicular to the straight line segment connecting $\mathbf{u, v}$? Kuldeep Singh. Linear Algebra: Step by Step
(2013). p 312
Why isn't $\mathbf{proj_u v}$ defined as the red vector below with $\mathbf{v} \perp \mathbf{d}?$ I already know, b... | It's so we can reconstruct the original vector easily from its projections. If we have an orthonormal basis $B=\{b_1,\dots, b_n\}$ of a vector space $V$, and any vector $v\in V$, then we have
$$v=\sum_{i=1}^n \operatorname{proj}_{b_i}(v).$$
Essentially, $\operatorname{proj}_uv$ is the "$u$-component" of $v$. For instan... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3771550",
"timestamp": "2023-03-29T00:00:00",
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What do the sum of the reciprocal of n squared make if n is a natural number? I’ve recently found out that $\sum_{n=1}^{\infty}\frac{1}{n^2+n}$ makes 1, since it becomes $\frac{1}{2}, \frac{2}{3}$ and so on. After then, I’ve became curious if I do the same thing with the reciprocal of n squared, or $$\sum_{n=1}^{\infty... | The sum is $\pi^2/6$. Euler first figured that out. It's no surprise and no disgrace that you didn't. See
https://en.wikipedia.org/wiki/Basel_problem .
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3771650",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Represent $f(x) - f(y)$ as an integral Description
I've come across the following transition in a textbook of Convex Optimisation. I couldn't figure out what's going on so that I'd appreciate if anyone hits me with any hint!
Problem
Suppose $x, y \in \mathbb{R}^n$ and $f$ be a $\beta$-smooth convex function on $\mathbb... | This is just the fundamental theorem of line integrals, which says for (sufficiently smooth) functions $f$,
$$f(x) - f(y) = \int_{C} \overrightarrow{\nabla f(r)} \cdot \vec{dr}$$
where $x$ and $y$ are the endpoints of $C$. Think of it as a generalization of the fundamental theorem calculus to higher dimensions, which r... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Four Number Theorem : Let $a$, $b$, $c$, $d$ be integers such that $ab = cd$. Let $a$, $b$, $c$, $d$ be integers such that $ab = cd$. Then there exist integers $x$, $y$, $z$, $w$ such that $xy = a$, $zw = b$, $xz = c$, $yw = d$.
My Progress: I tried playing with $\gcd(a,c)$, $\gcd(a,d)$,$\gcd(b,c)$, $\gcd(b,d)$.
But I ... | Firstly note that it is sufficient to prove the theorem when $a,b,c,d,x,y,z,w$ are all natural numbers. For if any of the given numbers is $0$ then the solution tuple $(x,y,z,w)$ is trivial and if there are negatives involved you can look for $x,y,z,w$ for $\lvert a \rvert,\lvert b \rvert, \lvert c \rvert, \lvert d \r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3771853",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Confusion of additive and multiplicative notations.
If gcd$(m,n)=1$, then $\phi(mn)=\phi(m)\phi(n)$.
My text book write that $f:\Bbb Z/_{(mn)}\Bbb Z\to\Bbb Z/_m\Bbb Z \times \Bbb Z/_n\Bbb Z$ by $f([a]_{mn})=([a]_m,[a]_n)$ is an isomorphism.
So far, I think he discusses the additive group.
But when he claims that $f(... | The first isomorphism refers to a ring isomorphism.
The second assertion refers to the mapping of the multiplicative groups of the involved rings. They are mapped accordingly.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3771972",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Simplify $\displaystyle\prod_{j=1}^{K}\exp(-2\pi\lambda_j(sp_j)^{2/\alpha}\int_0^{\infty}r\int_0^{\infty}e^{-t(1+r^{\alpha})}dtdr)$. An equation involving the poisson point process is formulated as:
$$\prod_{j=1}^{K}\exp(-2\pi\lambda_j(sp_j)^{2/\alpha}\int_0^{\infty}r\int_0^{\infty}e^{-t(1+r^{\alpha})}dtdr).$$
Some alg... | Focus on simplifying $$2\pi\int_0^{\infty}r\int_0^{\infty}e^{-t(1+r^{\alpha})}dtdr
=2\pi\int_0^{\infty}\frac r{1+r^\alpha}dr$$
Substitute $u=\frac1{1+r^\alpha}$ to get a Beta function form $$\frac{2\pi}{\alpha}\int_0^1(1-u)^{\frac2\alpha-1}u^\frac2\alpha du=\frac{2\pi}{\alpha}\Gamma\left(1-\frac2\alpha\right)\Gamma\lef... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3772088",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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A particular problem on series
Problem: Show that there exist $c>0$ such that for all $N\in \mathbb N$ we have $$
\sum_{n=N+1}^{\infty}\left(\sqrt{n+\frac{1}{n}}-\sqrt{n}\right)\le \frac{c}{\sqrt{N}}
$$
I have no clue how to solve this. All I know is this fact $$\int_0^1\left(\sum_{n\in \mathbb N}\frac{1}{
\sqrt{n^3+... | $$ \sqrt{n+\frac{1}{n}} -\sqrt n = \frac{1}{n\left(\sqrt{n+\frac{1}{n}} +\sqrt n\right)} \leq \frac{1}{2n^{3/2}} .$$
Now
$$ \sum_{n=N+1}^\infty n^{-3/2} \leq \int_N^\infty x^{-3/2} dx =\frac{ 2}{\sqrt N}$$
| {
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"source": "stackexchange",
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Find the asymptote and the upper bound for $1\big/ \ln\left(\frac{x+1}{x-1}\right)$ as $x\to\infty$. I need to show that $\left[\ln\left(\frac{x+1}{x-1}\right)\right]^{-1}$ asymptotically approaches to a line from below that I should determine, and hence, show that $\frac{x}{2} > \left[\ln\left(\frac{x+1}{x-1}\right)\r... | Fix $x > 1$ and write $\varphi(t) = \frac{1}{x+t}$. Since $\varphi_x$ is strictly convex, it follows from the Jensen's inequality that
\begin{align*}
\frac{1}{2} \log\left(\frac{x+1}{x-1}\right)
= \frac{1}{2} \int_{-1}^{1} \varphi_x(t) \, \mathrm{d}t
> \varphi_x \left( \frac{1}{2} \int_{-1}^{1} t \, \mathrm{d}t \right)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3772329",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Product of two Riemann surfaces $X$ with $H^1(X,T_X)In Buchdahl's paper Algebraic deformations of compact Kähler surfaces, the author made a remark that: the product of two Riemann surfaces of genus at least 5 satisfies the dimension of $H^1(X,T_X)$ < dimension of $H^2(X,\mathcal{O})$, but I can't see why, why the genu... | If $X = C \times D$ then $T_X = T_C \boxtimes \mathcal{O}_D \oplus \mathcal{O}_C \boxtimes T_D$ and by Kunneth formula
$$
h^1(X,T_X) = h^1(T_C)h^0(\mathcal{O}_D) + h^0(T_C)h^1(\mathcal{O}_D) + h^0(\mathcal{O}_C)h^1(T_D) + h^1(\mathcal{O}_C)h^0(T_D) = (3g(C) - 3) + 0 + (3g(D) - 3) + 0.
$$
Similarly,
$$
h^2(X,\mathcal{O}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3772478",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Prove $\exists (\varphi_1, \ldots, \varphi_m) \in \mathcal{L}(V, \mathbf{F}) : T(v) = \varphi_1(v)w_1 + \cdots + \varphi_m(v)w_m$, $\forall v \in V$
Suppose $T \in \mathcal{L}(V, W)$, and $(w_1, \ldots, w_m)$ is a basis of $\operatorname{range}(T)$. Prove that there exists $(\varphi_1, \ldots, \varphi_m) \in \mathcal{... | Here's a slightly different approach:
Given $v\in V$, let $\Phi(v)\in F^m$ be the unique $m$-tuple such that
$$
T(v)=(\Phi(v))_1w_1+\dots+(\Phi(v))_mw_m
$$
where we write $x_i$ for the $i$th entry of the tuple $x$. This definition makes sense because $w_1,\dots,w_m$ form a basis.
Let $\varphi_i(v):=(\Phi(v))_i$.
You ca... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3772558",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Solving the system $\sqrt{x} + y = 7$, $x + \sqrt{y} = 11$ I want to solve the following nonlinear system of algebraic equations. Indeed, I am curious about a step by step solution for pedagogical purposes. I am wondering if you can come up with anything. I tried but to no avail.
\begin{align*}
\sqrt{x} + y &= 7 \\
x +... | Rearrange both equations to isolate the term with the square root, then square both sides. This new system will likely add additional solutions, so we will need to check to make sure any result we get is really a solution.
Our new system is:
$$x=y^2-14y+49$$
$$y=x^2-22x+121$$
Substituting x from the first equation int... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3772635",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 7,
"answer_id": 4
} |
proposition of derivate Let $(X,{\tau})$ a topological space and $A$ a subset of $X$, show that:
$\operatorname{der}A=\{x\in \operatorname{cl}A\mid \operatorname{cl}A=\operatorname{cl}(A-\{x\})\}$
So I try to prove this. I suppose $x\in \operatorname{der}A$ by definition, for all $V\in\mathcal{N}(x)$, $V\cap(A-\{x\})\n... | You also need that $x \in \operatorname{cl}(A)$ iff for all $V \in \mathcal{N}(x)$, $V \cap A \neq \emptyset$, the closure of $A$ is the set of all adherent points of $A$.
So $x \in \operatorname{der}(A)$ means that $x$ is an adherent point of $A - \{x\} $, or $x \in \operatorname{cl}(A-\{x\})\subseteq \operatorname{cl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3772735",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Let $E\subset\mathbb{R^{n}}$ and $\hat{p}\in\mathbb{R^{n}}$, then all interior points of $E$ are limit points of $E$ as well. Let $E\subset\mathbb{R^{n}}$ and $\hat{p}\in\mathbb{R^{n}}$, then all interior points of $E$ are limit points of $E$ as well.
If $\hat{p}\in int(E)$ where $int(E)$ is the set containing all the ... | Hint: If $\hat{p}\in{\rm int}E$ then there exists $Ḅ̣_{r}(\hat{p})$ with
$Ḅ̣_{r}(\hat{p})\subset E$.
What do you add for the punctured ball
$Ḅ̣_{r}(\hat{p})\smallsetminus\{\hat{p}\}$?
What happens if
$(Ḅ̣_{r}(\hat{p})\smallsetminus\{\hat{p}\})\cap E=\varnothing?$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3772859",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Is there a closed form of $\sum_{n=0}^{\infty} \frac{(-1)^n}{(4n+1)!!}$? This may be an impossible problem. But I imagine it's worth asking still.
What is the closed form of the sum:
$$\sum_{n=0}^{\infty} \frac{(-1)^n}{(4n+1)!!}$$
Perhaps there isn't a closed form. Double factorials are WAY out of my comfort zone so ... | Too long for a comment.
Using the same approach as in @Xoque55's answer, we could go one step further and consider
$$f(x)=\sum_{n=0}^{\infty} \frac{(-1)^n}{(4n+1)!!}x^{4n}=\, _1F_2\left(1;\frac{3}{4},\frac{5}{4};-\frac{x^4}{16}\right)$$ which write
$$f(x)=\frac{\sqrt \pi}x \left(C\left(\frac{x}{\sqrt{\pi }}\right) \cos... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3772943",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
How I get the nth derivative of the function $y = e^x x^2$ I'm totally confused. Please anyone help me.
$y' = e^x x^2 + 2 e^xx$
$y''= e^x x^2 + 4 e^xx + 2e^x$
$y'''= e^x x^2 + 6 e^xx + 6 e^x$
next $y''''$ but I failed to get any pattern.
| We need to find an expression to $\frac{d^{n}y}{dx^{n}}$.
Observe that you can use $\underline{\text{product rule}}$ and $\underline{\text{chain rule}}$ in each calculation of the derivative to $y$. Indeed
$$\frac{dy}{dx}=e^{x}(x^{2}+2x+(1-1)(1))$$
$$\frac{d^{2}y}{dx^{2}}=e^{x}(x^{2}+2(2)x+(2-1)(2))$$
$$\frac{d^{3}y}{d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3773051",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 6,
"answer_id": 3
} |
Is $\operatorname{Aut}(D_{12})\simeq D_{12}$? Let $D_{12}$ be the dihedral group of order 12. Then
$$|\operatorname{Aut}(D_{12})|=6\phi(6)=12=|D_{12}|,$$
and the standard method of proof for
$$\operatorname{Aut}(D_6)\simeq D_{6}\qquad\mbox{and}\qquad \operatorname{Aut}(D_8)\simeq D_{8}$$
seems to also work for
$$\opera... | The article is wrong. First, the automorphism group is definitely $D_{12}$: here is Magma code that proves it.
> G:=DihedralGroup(6);
> A:=AutomorphismGroup(G);
> A:=PermutationGroup(A);
> IdentifyGroup(A);
<12, 4>
> IdentifyGroup(G);
<12, 4>
Second, even his Theorem A states that $\mathrm{Aut}(D_{12})\cong \mathrm{Ho... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3773139",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
For What positive values of x ; the below mentioned series is convergent and divergent? $$ \sum \frac{1}{x^{n}+x^{-n}}$$
My attempt
$$
\begin{aligned}
&\begin{aligned}
\therefore & u_{n+1}=\frac{x^{n+1}}{x^{2 n+2}+1} \\
\therefore & \frac{u_{n+1}}{u_{n}}=\frac{x^{n+1}}{x^{2 n+2}+1} \cdot \frac{x^{2 n}+1}{x^{n}} \\
& \f... | In your attempt you have shown that $\lim \sup \frac {u_{n+1}} {u_n} \leq x \frac {1+0} {x^{2}+0}=\frac 1 x <1$. Hence Ratio Test tells you that the series is convergent for $x>1$. But there is a bettter approach as shown below:
For $x>1$ it is dominated by $\sum \frac 1 {x^{n}}$ which is convergent.
For $0<x<1$ it is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3773204",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
RMM 2015 /P1: Does there exist an infinite sequence of positive integers $a_1, a_2, a_3, . . .$ Does there exist an infinite sequence of positive integers $a_1, a_2, a_3, . . .$ such that $a_m$ and $a_n$ are coprime if and only if $|m - n| = 1$?
My Progress:This is a very beautiful problem ! I think I have got a constr... | We'll do an inductive process, defining $a_{i,j}$ for integers $i \ge 1$ and $j \ge 0$.
Let $p_n$ be the $n$'th prime.
Initially, take $a_{1,1}= p_1 p_2 $, $a_{2,1} = 1$, $a_{n,1} = p_1$ if $n \ge 3$ is odd and $p_2$ if $n \ge 4$ is even. Note that $a_{n,1}$ and $a_{n+1,1}$ are coprime, and $a_{1,1}$ and $a_{n,1}$ are... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3773342",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Problem with the power of functions' set and number of discontinuity points I am considering functions $f:\mathbb{R}\rightarrow \mathbb{R}$ with property that $\forall_{r\in\mathbb{R}}$ exists a limit $\lim_{x\rightarrow r}f(x)$ (it doesn't have to be equal to $f(r)$). I have two problems.
The first with showing that ... | Whatever you indicate, it can be either a function with removable discontinuity or a continuous function.
Whenever $f$ have removable discontinuity, then $f$ can't be a monotone function,
For example, $f(x) =
\begin{cases}
2, & \text{if x=1} \\
x, & \text{otherwise}
\end{cases} $
And, how many are there such type func... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3773477",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Indefinite integral of $\sin^8(x)$ Suppose we have the following function:
$$\sin^8(x)$$
We have to find its anti-derivative
To find the indefinite integral of $\sin^4(x)$, I converted everything to $\cos(2x)$ and $\cos(4x)$ and then integrated. However this method wont be suitable to find the indefinite integral $\si... | The development through the double angle formulae is not so long, let me show.
\begin{align}
\sin^8x
&=(\sin^2x)^4=\\
&=\left(\frac{1-\cos2x}{2}\right)^4=\\
&=\frac{1}{16}(1-4\cos2x+6\cos^22x-4\cos^32x+\cos^42x)=\\
&=\frac{1}{16}[1-4\cos2x+3(1+\cos4x)-4\cos2x(1-\sin^22x)+(\cos^22x)^2]=\\
&=\frac{1}{16}\left[1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3773595",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 1
} |
Determine when $\sum_1^\infty \frac{(2n)!x^n}{n(n!)^2}$ converges. Determine when $\sum_1^\infty \frac{(2n)!x^n}{n(n!)^2}$ converges.
By ratio test, when $|x|<1/4$, the sum converges, $|x|>1/4$ diverges. But I'm not sure about $|x|=1/4$.
By Stirling's approximation $\frac{(2n)!(1/4)^n}{n(n!)^2}\sim\frac{4^n}{n{4^n}\sqr... | Let's study what happens when $x=1/4$.
Define $a_n:=\frac{(2n)!x^n}{n(n!)^2} $, $b_n:=\dfrac{1}{n^{5/4}}$ and observe the following growth rates:
$$ \dfrac{a_{n+1}}{a_n}
= \frac{(2n+2)!x^{n+1}}{(n+1)((n+1)!)^2} \frac{n(n!)^2}{(2n)!x^n}
= \dfrac{(2n+2)(2n+1)n}{4(n+1)^3}
= \dfrac{(2n+1)n}{2(n+1)^2}=\big(1-\dfrac{1}{n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3773681",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Using generating function to solve non-homogenous recurrence relation The given recurrence relation is:
$$
a_{n} + 2a_{n-2} = 2n + 3
$$
with initial conditions: $$ a_{0}=3$$ $$a_{1}=5$$
I know
$$G(x) = a_0x^0 + a_1x^1 + \sum_{n=2}^{\infty} a_nx^n $$
and
$$ a_{n} = -2a_{n-2} + 2n + 3$$
Therefore:
$$G(x) = 3 + 5x + \s... | $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\new... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3773771",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Inequality in a Sobolev space In the Sobolev space $H^2(\mathbb{R}^2)$ is it true that there exists a constant $c>0$ such that
$$\|u\|_{L^2(\mathbb{R}^2)}\le c\|\nabla u\|_{L^2(\mathbb{R}^2)}\;,\quad\quad \text{for all}\quad u\in H^2(\mathbb{R}^2)?$$
Thank you in advance.
| So this is called the Poincaré inequality. It is true on a bounded domain. When you are in $\mathbb{R}^d$ and not in a bounded domain, you need some additional weights (for example it works with Gaussian weights, or sufficiently decaying weights).
In the form you are asking, it is false. indeed, take $\langle x\rangle ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3773864",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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How do I create an offset shape that is a specific distance from a given circle, in the direction of the origin? I'm an amateur engineer, working on a CAD design - but sadly, I'm not a mathematician.
In other words, this question might sound like homework, but it's not, I promise.
I have an existing circle, which has a... | The equation of the circle is $(x,y)=(r\cos\theta,a+r\sin\theta)$. The question is for the curve $s(x,y)$ such that $(1-s)|(x,y)|=d$. Solving this gives $s=1-\frac{d}{\sqrt{x^2+y^2}}$.
Hence the curve is given by the pseudo-code:
r=23.5; a=15;
x(t):=r*cos(t);
y(t):=a+r*sin(t);
s(t):=1-d/sqrt(x(t)^2+y(t)^2);
plot((s(t)*... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3773955",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Right exactness of quotienting out the maximal divisible subgroup
For every abelian groups $G$ let $\mathrm{d}G$ be its maximal divisible subgroup.
Then $G \mapsto G/\mathrm{d}G$ is a right exact functor $\mathbf{Ab} \to \mathbf{Ab}$.
Let
$$
0 \to G \xrightarrow{i} H \xrightarrow{p} K \to 0
$$
be an exact sequenc... | Consider the short exact sequence: $$0\to\mathbb{Z}^{\mathbb{N}_{>0}}\stackrel i \to \mathbb{Z}^{\mathbb{N}_{>0}}\stackrel p\to \mathbb{Q}\to 0,$$
where $i(f_n)=(n+1)e_{n+1}-e_n$ and $p(e_n)=\frac1{n!}$ for $n\geq 1$.
Applying the functor results in the sequence:
$$0\to\mathbb{Z}^{\mathbb{N}_{>0}}\stackrel i \to \mathb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3774113",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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$n-$circular arrangement problem Find the number of ways to arrange $n$ people in a circle so that $3$ people are separated.
My approach: The number of ways to arrange $n$ people in a circle is $(n - 1)!$. If the $3$ people are together, the number of arrangement is $(n - 3)!$. The $3$ people can rearrange themselves ... | Comment of @Christian Blatter might be helpful to you in counting the missing cases. However, here's an alternate method.
Let the three people be $P_1,\ P_2,\ P_3$ and the number of people between $P_1P_2,\ P_2P_3,\ P_3P_1$ be $x_1,\ x_2,\ x_3$ respectively. Now, we've to find number of positive integral solutions of t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3774236",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Show that the kernel ker(B) is a vector subspace of the domain. I want to know how to show that $\ker(B)$ is a vector subspace of the domain.
| I'll sketch how to obtain the reduced row echelon form:
\begin{align}
&\phantom{{}\rightsquigarrow{}} \begin{pmatrix}
3 & -3 & 1 & 5 & 1 & 5 & 5\\
1 & -1 & 1 & 1 & 1 & 3 & -1\\
2 & -2 & 1 & 3 & 0 & 5 & 2\\
2 & -2 & 0 & 4 & 0 & 2 & 1
\end{pmatrix}\rightsquigarrow
\begin{pmatrix}
1 & -1 & 1 & 1 & 1 & 3 & -1\\
3 & -3 & ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3774357",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
If $y = \frac{2}{5}+\frac{1\cdot3}{2!} \left(\frac{2}{5}\right)^2+\frac{1\cdot3\cdot5}{3!} \left(\frac{2}{5}\right)^3+\cdots$, find $y^2+2y$
If $$y = \frac{2}{5}+\frac{1\cdot3}{2!} \left(\frac{2}{5}\right)^2+\frac{1\cdot3\cdot5}{3!} \left(\frac{2}{5}\right)^3+\cdots$$ what is $y^2+2y$?
Attempt:
We know that for negat... | Substitute $x=\frac{2}{5}$:
$$y=\sum_{i=1}^\infty\frac{x^i}{i!}(\prod_{k=1}^{i}(2k-1))$$
$$y=\sum_{i=1}^\infty\frac{x^i}{i!}((2i-1)!!)$$
For $|x|<\frac{1}{2}$, this is the Taylor series for $\frac{1}{\sqrt{1-2x}}-1$
$$y=\frac{1}{\sqrt{1-2x}}-1$$
$$y=\frac{1}{\sqrt{1-0.8}}-1$$
$$y=\frac{1}{\sqrt{0.2}}-1$$
$$y=\sqrt{5}-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3774460",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Showing whether an ideal in $\mathbb{Z}[x,y]$ is prime. The ideal $(1+x^2,1+y^2)$ is prime in $\mathbb{Z}[x,y]$?
I have this:
Analogously to $\mathbb{Z}[x]/(1+x^2)\simeq \mathbb{Z}[i]$, $\mathbb{Z}[x,y]/(1+x^2,1+y^2)\simeq \mathbb{Z}[i]\times \mathbb{Z}[i]$ and $\mathbb{Z}[i]\times \mathbb{Z}[i]$ is not a integral dom... | You need a completer argument why $$\mathbb{Z}[x,y]/(1+x^2,1+y^2)\simeq \mathbb{Z}[i]\times \mathbb{Z}[i]$$
I’m not even sure it is true.
I think the quotient ring is $\mathbb{Z}[i]\otimes_{\mathbb Z}\mathbb{Z}[i].$
Then the zero divisors are $$(i\otimes 1 +1\otimes i)(i\otimes 1-1\otimes i)=0.$$
You are correct, thou... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3774679",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
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How do we solve the equation $2^{x^2-3}=x^{-1/3}$ algebraically? This question was from Khan Academy and, even though Sal solved it through graphing, I want to know how it can be solved algebraically.
Here are the steps that I have tried:
$2^{x^2-3}=x^{-1/3}$
$2^{{(x^2-3)}^{-3}}=x^{{(-1/3)}^{-3}}$
$2^{-3x^2+9}=x$
$\log... | There is no explicit solution if you cannot use Lambert function and some numerical method will be required.
Solving $$2^{x^2-3}=x^{-1/3}$$ is just the same as finding the zero of function
$$f(x)=\log \left(2^{x^2-3} \sqrt[3]{x}\right)$$ If you plot it, you will see that the root is close to $1.65$ which is close to $\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3774766",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Inequality involving size of random variable I’m following my professor’s notes, and I became confused by an inequality he used without justification, but I cannot see why it is true. Any help is appreciated.
Let $\{T_t\}_t$ be a sequence of iid, positive, integer-valued random variables. Suppose $$P\{T \ge x \} = \The... | Say $ a/\sqrt{x} \le P(T \ge x) \le b/\sqrt{x}$ for $x$ sufficiently large.
For (1):
$$ P\left(\max_{1\le i \le t} T_i < c t^2\right) = P(T < c t^2)^t $$
When $c > 0$, for $t$ sufficiently large this is between $(1 - a/(\sqrt{c} t))^t$
and $(1 - b/(\sqrt{c} t))^t$, which as $t \to \infty$ go to $\exp(-a/\sqrt{c})$ and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3774979",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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In triangle $\triangle ABC$, angle $\angle B$ is equal to $60^\circ$; bisectors $AD$ and $CE$ intersect at point $O$. Prove that $OD=OE$. In triangle $\triangle ABC$, angle $\angle B$ is equal to $60^{\circ}$; bisectors $AD$ and $CE$ intersect at point $O$. Prove that $OD=OE$.
So I've already made a diagram(it is attac... | Note that O is the incenter of Triangle ABC.
Thus, perpendiculars from O to sides AB and BC have equal length. Let them be G and H respectively. Can you prove that $\triangle$ OGE and $\triangle$ OHD are congruent?
Hint : Use the fact that $\angle$ AOC is $120^o$ and also the fact that $\angle$ GOH is also $120^o$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3775136",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Series expansion for real number Suppose c is a real number with 0 < c < 1. I am interested in the
following series.
c = 1/n1 + 1/n2 + 1/n3 + 1/n4 + .....
Where n1,n2, are positive integers chose to me as small as possible. For
example
4/5 = 1/2 + 1/4 + 1/20
e-2 = 1/2 + 1/5 + 1/55 + 1/9999 + ....
By the way this... | This general type of representation (sum of fractions with numerator 1 and no repeat denominators) is called Egyptian fractions. You'll find a lot of further facts in the cited page. Knowing what to search for will net you lots of further tidbits.
The name is because pharaonic Egyptians used this (awkward to operate) w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3775279",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Equation (3.89) seems wrong in Bishop pattern recognition & machine learning book In Bishop's pattern recognition & machine learning book, I seem to have found a serious mistake in an math equation; serious because all subsequent arguments rely on it.
It is the eq. (3.89) on page 168:
$$
0 = \frac{M}{2\alpha} -\frac{1}... | You are not making a mistake, you just need to go one step further. Fist, note that $\mathbf{m}_{N}=\beta \mathbf{A}^{-1} \mathbf{\Phi}^{\mathrm{T}} \mathbf{t}$ with $\mathbf{A} = \alpha I + \beta \boldsymbol{\Phi}^{T}\boldsymbol{\Phi}$. Having that in mind we can start by working your expression out
$$
\frac{\partial ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3775407",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Show that Rank $(A)$ = Rank$(A^2)$ $\iff \exists$ B invertible s.t. $A^2= B A$. Given $n \times n$ matrix $A$, not necessarily invertible, what invertible matrix $B$ solves $A^2= B A$?
$B$ should be $A$ but that is what not being said in the claim above. Please help. Thanks in advance.
| We have yet to see that the converse is true, so here is a bit about that.
Consider an $n \times n$ matrix over the field $k.$ Given an invertible matrix $B$ such that $A^2 = BA,$ we claim that $\ker(A^2) = \ker(BA) = \ker(A).$ Certainly, we have that $\ker(A) \subseteq \ker(A^2),$ as any vector $v$ such that $Av = 0$ ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
$H^1$-conforming approximation for elliptic PDE with discontinuous coefficient? everyone
Suppose I want to solve the diffusion equation $$-\nabla\cdot a \nabla u=f, \\u=0 \text{ on } \partial \Omega,$$ $f \in L^2(\Omega)$, $\partial \Omega$ is smooth.
I use the standard node-based linear elements on a tetrahedral mesh.... | It is quite standard to have $a$ piecewise constant, or more generally a piecewise constant diffusion matrix. Here we consider $a:\Omega\to\mathbb{R}$, but shall require additional regularity.
In the stated problem, you have already assumed that the gradient exists. So your question actually boils down to whether or no... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3775684",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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} |
Prove $\int_0^{\infty} \frac{\arctan{(x)}}{x} \ln{\left(\frac{1+x^2}{{(1-x)}^2}\right)} \; \mathrm{d}x = \frac{3\pi^3}{16}$ Prove that $$\int_0^{\infty} \frac{\arctan{(x)}}{x} \ln{\left(\frac{1+x^2}{{(1-x)}^2}\right)} \; \mathrm{d}x = \frac{3\pi^3}{16}$$
This is not a duplicate of this post, the bounds are different an... | Changing the bounds makes the integral way simpler, because after letting $x\to \frac{1}{x}$ we can get rid of that $\arctan x$.
$$I=\int_0^{\infty} \frac{\arctan x}{x} \ln\left(\frac{1+x^2}{{(1-x)}^2}\right)dx\overset{x\to \frac{1}{x}}=\int_0^\infty \frac{\arctan \left(\frac{1}{x}\right)}{x}\ln\left(\frac{1+x^2}{(1-x)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3775808",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Problem with extension of a continous function
Let $X$ a first-countable Topological Space, let $Y$ an Hausdorff Topological Space, let $A\subset X$ a subset ot $X$ and let $f:A\rightarrow Y$ a continous function.
Prove that, if there is an extension
$$\overline{f} :\overline{A}\rightarrow Y$$
$\overline{f}$ is solely... | Your argument works. You may need to elaborate a bit on
Then, $Y$ is a Hausdorff space so, $Z$ is closed in $X$
and it may be preferable to write $Z$ is closed in $\overline{A}$ there. Whether you need to elaborate or not depends on what properties of Hausdorff spaces and continuous maps can be assumed as generally k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3775924",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Calculate $61^{61} \pmod9$ I have a solution in front of me, but several parts do not quite make sense to me.
Since $61 \equiv −2 \pmod9$, we have
$$61^{61} \equiv (−2)^{61}\equiv −2 \cdot 8^{60}\pmod{9}$$
First question: how do we get from $(-2)^{61}$ is congruent to $(-2)\cdot 8^{60}$?
$$-2\cdot 8^{60}\equiv −2 · (−1... | $61^{61}\equiv(-2)^{61}\equiv(-2)(8)^{20}\equiv(-2)(-1)^{20}\equiv-2\equiv7\pmod9.$
Alternatively, by Euler's theorem, $a^6\equiv1\pmod9$ if $\gcd(a,9)=1$,
so $61^{61}\equiv(-2)^{61}\equiv(-2)^{60}(-2)\equiv((-2)^6)^{10}(-2)\equiv7\pmod9.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3776016",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Proving a result for $\prod_{k=0}^{\infty}\Bigl(1-\frac{4}{(4k+a)^2}\Bigr)$ $$\prod_{k=0}^{\infty}\Bigl(1-\frac{4}{(4k+a)^2}\Bigr)=\frac{(a^2-4)\Gamma^2\bigl(\frac{a+4}{4}\bigr)}{a^2\Gamma\bigl(\frac{a+2}{4}\bigr)\Gamma\bigl(\frac{a+6}{4}\bigr)}$$
According to WA. I attempted using
$$\prod_{k=0}^{\infty}\Bigl(1-\frac{x... | It isn't pretty, but we can prove this by working backwards using Euler's product definition of the gamma function (seen here):
$$
\Gamma(x) = \lim_{n\to\infty} n!(n+1)^x \prod_{k=0}^n (x+k)^{-1}.
$$
When we substitute that in for the right hand side, we get
$$
\begin{align}
\frac{(a^2-4)\Gamma^2\bigl(\frac{a+4}{4}\big... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3776147",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Solve Bernouilli differential equation I think that my question is dumb... but I am stuck in a step.
I need to solve this ode: $$\rho' = \rho(1+\rho^2)$$
I made the variable change $u:=\rho^{-2}$. Then, separating variables and replace $u$ by $\rho^{-2}$, i get $\rho=\frac{1}{\sqrt{e^{-2t}-1}}$.
But, when i check the s... | EDIT
Let $y = \rho$ for writing convenience. To begin with, notice that
\begin{align*}
y' = y(1+y^{2}) & \Longleftrightarrow \frac{y'}{y(1+y^{2})} = 1
\end{align*}
Then we have that
\begin{align*}
\frac{1}{y(1+y^{2})} = \frac{(1 + y^{2}) - y^{2}}{y(1+y^{2})} = \frac{1}{y} - \frac{y}{1+y^{2}}
\end{align*}
Finally, we ob... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
} |
If $A\cap B^\complement=\emptyset$ then $A\cap B=A$ I need help with this excercise.
If $A\cap B^\complement=\emptyset$ then $A\cap B=A$
I try
$$A\cap B^\complement =\emptyset$$
$$(A\cap B^\complement)\cap B =\emptyset \cap B$$
$$A\cap (B^\complement\cap B )= B$$
$$A=B$$
Is this reasoning correct?
| No, your reasoning leads nowhere because you get $\emptyset=\emptyset$. The property you want is a direct consequence of
$$
A=(A\cap B)\cup(A\cap B^c).
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3776324",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Does weak continuity imply continuity? I have come across the following excerpt from a mathematical Statistics book:
where $H$ and $J$ are Hilbert spaces and $H^{\star}$ is the dual space. For me, the statement after Definition 10 is unconvincing and I cannot, in general, show that weak continuity implies continuity. ... | 'Clearly' should not have been there but the result is true. This require the so-called Closed Graph Theorem (CGT). If $x_n \to x$ and $f(x_n) \to y$ in the norm then $f(x_n) \to f(x)$ weakly and it converges to $y$ in the norm (hence also weakly) and this implies $y=f(x)$. By CGT it follows that $f$ is continuous.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3776442",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Find what is span of 2 linearly independent vectors is I have been trying assignment questions of linear algebra and I am unable to solve this particular question
Let $x=\left(x_{1}, x_{2}, x_{3}\right), y=\left(y_{1}, y_{2}, y_{3}\right) \in \mathbb{R}^{3}$
be linearly independent. Let $\delta_{1}=x_{2} y_{3}-y_{2} x... | Given, $x=(x_1,x_2,x_3), y=(y_1,y_2,y_3) \in \mathbb{R}^3 $ are linearly independent.
Also, given, $V$ is the span of that two linearly independent vectors $x,y\in \mathbb{R}^3 $
It means clearly that each $(u,v,w)\in V $ can be uniquely spanned by the linearly independent vectors $x,y\in \mathbb{R}^3 $
That means for ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3776606",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How do I solve the following system of equations by using the Gauss-Jordan method? $$
\left\{\begin{array}{rcrcrcrcr}
x & - & 2y & + & 3z & - & 4w & = & 10
\\
2x & - & 3y & + & 4z & - & 5w & = & 18
\\
3x & - & 4y & + & 5z & - & 6w & = & 26
\\
4x & - & 5y & + & 6z & - & 7w & = & 9
\end{array}\right.
$$
Tried to solve t... | A whole line filled with 0s can be eliminated (it corresponds to equation 0=0).
The remaining system can be transformed into a square matrix if you pass to the other side the columns corresponding to $z$ and $w$.
1 0 -1 2|6
0 1 -2 3|-2
corresponds to
$x=6+\lambda-2\mu$
and $y=-2+2\lambda-3\mu$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3776711",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Approximation of open subsets of $\Bbb R^2$ by compact sets. I am having difficulty to prove one of the regularity properties of Lebesgue measure. Here it is $:$
Let $(\Bbb R^2, \mathcal L_{\Bbb R^2}, \lambda_{\Bbb R^2})$ be the Lebesgue measure space on $\Bbb R^2.$ Then prove that $\lambda_{\Bbb R^2} (U) = \sup \left ... | I proceed along the same lines as Greg Martin pointed out in his comment above. Here's my answer.
Let $S : = \left \{\lambda_{\Bbb R^2} (K)\ |\ K\ \text {is compact},\ K \subseteq U \right \}.$ We need to prove that $\lambda_{\Bbb R^2} (U) = \sup S.$
Since $U$ is open, $U$ is a neighbourhood of each of it's points. So ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3776800",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Is a locally compact Hausdorff quotient of a locally compact $\sigma$-compact first countable Hausdorff space always Frechet-Urysohn? This question follows on from a previous one, which has been answered in the negative: Is a locally compact Hausdorff quotient of a locally compact $\sigma$-compact first countable Hausd... | I describe an example here including the argument why the quotient space (there called $Y$ is not Fréchet-Urysohn (namely the map $q$ is not hereditarily quotient). The space we take a quotient of is a $\sigma$-compact and locally compact subspace of $\Bbb R$, so fits the bill.
The Arens space $S_2$ (which is Hausdorff... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3776926",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What is the difference between stochastic process and random variable? I am having a hard time grasping the core difference between a random variable and a stochastic process.
*
*A random variable assigns a number to every outcome of an experiment.
*A random process assigns a function of time to every outcome of an ... | Given probability space $(\Omega, \mathfrak{B}, P)$ random variable is measurable map
$$X:\Omega \to \mathbb{R} $$
while random (i.e. stochastic) process is family of random variables
$$X:\Omega \times T \to \mathbb{R}$$
where under $T$ often is considered as time.
On example you can understand it so: random variable r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3777027",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 0
} |
Set Operation $D = (A \times B) - (B \times C)$ I want to find
$$
D = (A \times B) - (B \times C)
$$
where
$$
A = \{x.y,z\}, \ B = \{1,2\}, \ C = \{x,z\}
$$
So far I have computed
$$
A \times B = \{(x,1), (x,2), (y,1),(y,2),(z,1),(z,2)\}
$$
and
$$
B \times C = \{(1,x),(1,z),(2,x),(2,z)\}
$$
Hence
$$
D = \{(x,1), (x,... | Because no element of $B \times C$ is an element of $A \times B$, the set $D$ is just $A \times B$. Remember: $X - Y = \{ x \in X : x \notin Y \}$ so if every element of $X$ is not an element of $Y$ then $x \notin Y$ is always satisfied so $X - Y = \{x \in X\} = X$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3777129",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Inverse image of anhilator ideals Let $f$ be a ring homomorphism from $R\rightarrow S$ and $J$ be annihilator of some ideal in $S$. Under what conditions on $R$ and kernel of $f$ , $f^{-1} (J) $ is annihilator of some ideal of $R$.
This question seems hard to me although its elementary and it gives me hard time in solv... | Let $f:R\to S$ be a ring homomorphism and $I$ an ideal of $S$ with annihilator $J$. Then $f^{-1}(I)$ is an ideal. If $j\in f^{-1}(J)$ and $a\in f^{-1}(I)$, then $f(ja)=f(j)f(a)=0$. So for $j$ to annihilate every element of $f^{-1}(I)$ it is necessary that the kernel of $f$ intersects $jf^{-1}(I)$ trivially. Therefore, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3777295",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Proving that sequence is in $\ell^\infty$ Let $(y_k)_{k\in\mathbb{N}}$ be a sequence in $\mathbb{K}$ with the property that for all sequences $(x_k)_{k\in\mathbb{N}}\in\ell^1$ the sequence $\sum_{k=1}^{n} x_k y_k$ has a limit in $\mathbb{K}$ for $n\to\infty$. Prove that $y\in \ell^\infty$.
I tried to define an operator... | Suppose the sequence $(y_n)_{n\in\mathbb{N}}$ to be unbounded.
Then for each $n>0$, there is $k_n$ such that $|y_{k_n}|\ge n$. Without loss of generality we can replace the original $y_n$ by this subsequence.
Let $x_n:=\frac{\overline{y}_n}{|y_n|n^2}$; the $n^2$ is introduced so $(x_n)\in\ell^1$. Then
$$
\sum_nx_ny_n=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3777388",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Is logical equivalence itself a proposition? I understand that the biconditional $P \leftrightarrow Q$ is a hypothesis that may be true or false depending on the truth values of P and Q.
Furthermore, I understand that the logical equivalence $P \Leftrightarrow Q$ is the assertion that the biconditional is a tautology, ... | It is an equivalence in a metalanguage, which loosely speaking is a "higher form" of equality. Basically saying that $P\Leftrightarrow Q$ says that "$P$ is true if and only if $Q$ is true", where being true is something derivable from the rules of propositional logic. This is different from saying "$P\leftrightarrow Q$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3777466",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How can a matrix represent a relation when a relation is not a function? A matrix can represent a relation (https://en.wikipedia.org/wiki/Logical_matrix). How is this so? Since matrices are a representation of linear maps, does this mean that a linear map is not necessarily a function?
| Don’t confuse the representation with the thing being represented. A PNG image might show a portrait of person, or a rendered math equation. Both are grids of pixels but that doesn’t mean that human faces are types of math equations (or vice versa).
Similarly a matrix is just a rectangular grid of numbers. Linear maps ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3777627",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Calculate the limit $\lim_{n\to\infty} \left(\prod_{k=1}^{n}(1+\frac{k}{n})\right)^{\frac{1}{n}}$ I have to calculate this limit $$ \lim_{n\to\infty} \left(\prod_{k=1}^{n}\left(1+\frac{k}{n}\right)\right)^{\frac{1}{n}}$$
I tried it just first to calculate the limit inside the product, but I think I got the answer 1. A... | As an alternative, we have that
$$\left(\prod_{k=1}^{n}\left(1+\frac{k}{n}\right)\right)^{\frac{1}{n}}=e^{\frac{\sum_{k=1}^{n} \log\left(1+\frac{k}{n}\right) }{n}}=e^{\sum_{k=1}^{n} \left(\frac{k}{n^2}-\frac12\frac{k^2}{n^3}+\frac13\frac{k^3}{n^4}+\ldots\right) }\to \frac4e$$
indeed by Faulhaber's formula
$$\sum_{k=1}^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3777859",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Multinomial coefficient of a sequence in specific form I came across a question which asked me to find the coefficient of $x^{2n}$ in the following polynomial:
$$(\sum\limits_{i=0}^{n-1} x^i )^{2n+1}$$
My approach was to isolate every term, i.e. if we choose $x^2$ , n times and 1 n times again, we get a part of the coe... | I would view it from the formal-power-series angle: $$\left(\sum_{k=0}^{n-1}x^k\right)^{2n+1}=\left(\frac{1-x^n}{1-x}\right)^{2n+1}=\underbrace{\left(\sum_{k=0}^{2n+1}(-1)^k\binom{2n+1}{k}x^{nk}\right)}_{=(1-x^n)^{2n+1}}\cdot\underbrace{\left(\sum_{k=0}^\infty\binom{2n+k}{k}x^k\right)}_{=(1-x)^{-2n-1}}$$ (both are inst... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3777940",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
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