Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
What is a matrix $A$ satisfying $A^m=I$ called? We have matrices that are idempotent $A^2=A$ or nilpotent $A^m=0$ for some $m$.
Question: What is a square matrix $A$ called for which $A^m=I$ for some integer $m$, where $I$ is the identity matrix?
Examples are rotations by $2\pi/m$.
Any reference would be appreciated!
| Such matrix $A$ is a matrix of finite order. You can find a classification in this article for matrices over $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{Q}$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why is the surface area of a sphere not $ 2 ( \pi r)^2 $? I was trying to derive the formula for the surface area of a sphere and thought of deriving it this way. If we have a circle with radius $r$ and we rotate it along its center by $180$ degrees, the circumference of the circle would cover each part of the sphere o... | As in my comment . . .
In your rotation, the distance traveled by a point is variable. Only two points travel a distance equal to a half circumference. The rest travel less.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3684805",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If $f(x)>0$ near $x_0$ and $\lim_{x\to x_0}f(x)$ exists, then why is it always $\lim_{x\to x_0}f(x)\geq0$?
If $f(x)>0$ near $x_0$ and $\lim_{x\to x_0}f(x)$ exists, then why is it always $\lim_{x\to x_0}f(x)\geq0$?
The basic limit theorems, state that If $\lim_{x \to x_0}f(x)>0$, then $f(x)>0$, near $x_0$. Why does th... | Consider the function $f:\mathbb{R}\to\mathbb{R}$ given by $f(x)=|x|.$ Then $f(x)>0$ for all $x \neq 0.$ But $\lim_{x\to 0} f(x)=0.$
However, if $\lim_{x\to x_0}=L>0,$ then corresponding to $\frac{L}2>0,$ there exists $\delta >0$ such that $|f(x)-L|<\frac{L}{2}$ whenever $|x-x_0| <\delta.$ So near $x_0,$ we have $f(x)>... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3684934",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Construct a circle tangent to sides $BC$ and $CD$ and s.t. its meetings with the diagonal $BD$ are tangent points from tangents draw from point $A$ Given square $ABCD$
I want to construct (with ruler and compass) the circle in the interior of the square such that it is tangent to sides $BC$ and $CD$ and such that it's ... | Let center $O$ of the circle lie on diagonal $\overline{AB}$ with midpoint $M$, and define $a:=|OA|$, $b:=|OB|$. Let the circle meet the other diagonal at $R$, and define $r:=|OR|$; note that $r=b/\sqrt{2}$.
$$\begin{align}
\underbrace{\frac{|OR|}{|OA|}=\frac{|OM|}{|OR|}}_{\triangle ORA\sim\triangle OMR} &\quad\to\qua... | {
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"url": "https://math.stackexchange.com/questions/3685145",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
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$\int x^{dx}-1$ If you go to Flammable Maths's YouTube channel and scroll through some of his videos you see him solving the following integral:
$$\int x^{dx}-1$$
he explains that this is a Product integral. My questions are the following:
1 - What is the geometric meaning of a product integral?
2 - does it make sense... | $\int x^{dx}-1$ = $\int \frac{x^{dx}-1}{{dx}} {dx}$ = $\int (\lim_{h \to 0}\frac{x^h-1}{h}) {dx}$ = $\int \ln x {dx}$ = $x\ln x-x+const.$
$ \therefore \int x^{dx}-1 = x \ln x - x + const.$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Find good bounds for the inverse of a function The general case:
Let $f : A\subset \mathbb R \to B = f(A)\subset \mathbb R$ be a $\mathcal C^\infty$-diffeormorphism.
Find good polynomial bounds for $f$ at any order (i.e. $P,Q \in \mathbb R_n[X]$ such as $\forall x \in B, P(x)\leq f(x) \leq Q(x)$ for every $n$).
What I ... | Concerning your special case, $\lim_{s\to\infty} f(s) = 0$. Also there is a point $s_0$ with $f$ decreasing on $[s_0, +\infty)$. Exactly how low $s_0$ can be taken is difficult to determine, but because of the $\sin \frac{2\alpha}s$ factor, it is going to be somewhere around $\frac {4\alpha}\pi$, well above the asympto... | {
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"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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The Number of Hyperplanes Intersecting a Unit Hypercube Prove that the number of hyperplanes such that
$$c_1x_1 + c_2 x_2 + ... + c_n x_n = 0, \pm 1, \pm 2, \pm 3, ...$$
which intersect the unit $n$-cube, $0< x_i < 1,$ is at most
$$|c_1| + |c_2| + ... + |c_n|.$$
I started out by plotting small values of $c_i$ in the ... | First assume that each $c_i\ge 0$, and set $C:=c_1+\dots+c_n$.
Consider the family of hyperplanes
$$H_t=\{(x_1,\dots, x_n)\, :\, c_1x_1+\dots+c_nx_n= t\} $$
for all $t\in\Bbb R$. Note that these hyperplanes are all parallel and have the normal vector $(c_1,\dots, c_n)$.
Geometrically it's obvious, but you can also der... | {
"language": "en",
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"source": "stackexchange",
"question_score": "4",
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What are the eigenvalues of $T(A) = A+ (A)_{2,2}I$, where $T\colon M_2(\mathbb{C})\to M_2(\mathbb{C})$ is a linear operator let $T$ be linear transformation $T:M_2(\mathbb{C})\to M_2(\mathbb{C})$.
$$T(A) = A+ (A)_{2,2}I$$
I need to find its eigenvalues, does it not depend on the value of $(A)_{2,2}$?
I dont understand ... | One approach here is to select a basis, find the $4 \times 4$ matrix of the linear transformation relative to this basis, then find the eigenvalues and eigenvectors of this matrix in the usual way. In this case, however, I prefer to work directly from the definition of eigenvalues and eigenvectors.
Note that $\lambda$ ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
} |
proving that $f_n$ doesn't uniformly converge I am trying to prove that $$fn(x)=(nx)/(1+n^2x^4)$$ doesn't uniformly converge in [0,1]
For that I'm looking for K a real positive fixed number, m,n $\in \mathbb{N}$ which achieve the following inequality:
$$|nx/(1+n^2x^4) - mx/(1+m^2x^4)| \ge K. $$
Why I'm doing this?
wel... | It is easier to observe that
for $n>0$
$$\sup_{x\in \Bbb R}|f_n(x)-0|\ge $$
$$f_n(\frac{1}{\sqrt{n}})=\frac{\sqrt{n}}{2}$$
thus
$$\lim_{n\to +\infty}\sup_{x\in\Bbb R}|f_n(x)-0|=+\infty$$
The convergence is not uniform.
| {
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"question_score": "1",
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Unit vectors and quaternions This might be a dumb question but recently I came to know about the quaternion number system.
I can't stop wondering, "Are they related to unit vectors in any way? They have similar notations and both seem to be related to Spatial dimensions."
| As Don Thousand explains, you do need a bit of Linear Algebra/Abstract Algebra to fully appreciate the relationships. I would also add some elementary Complex Analysis (just the operations and basics) to also appreciate/understand. But for an application of quaternions and their vector connections, with a bit of the Ma... | {
"language": "en",
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Sum of digits of sum of digits of powers of 12345 Sum of digits of $12345$ is $1+2+3+4+5=15$. The sum of digits of sum of digits is $1+5=6$.
I have plotted the sum of digits of powers of $12345$ with blue dots (x-axis is the power).
As the average digit is $4.5$, we can see the roughly linear rise with slope $log_{10}(... | As $12345$ is a multiple of $3$, the powers of $12345$, excluding itself, will be multiples of $9$. By the divisibility rule of $9$ the sum of the digits, and the sum of the sum of the digits will also be divisible by $9$. (That is why the first dot alone was an exception).
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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Interpretation of the notation $x = (x_1,x_2)\in \{0,1\}^2$? I have a few questions regarding the following notation:
$$
x = (x_1,x_2)\in \{0,1\}^2
$$
Question 1:
Is the following correct?
$\{0,1\}^2$ is the Cartesian product of the 2 sets $\{0,1\}$ and $\{0,1\}$, i.e.
\begin{align}
\{0,1\}^2 &= \{0,1\} \times \{0,1\... | Yes, yes and yes.
In general, the notation $A^n$, for a set $A$ and natural number $n$, means
$$
\underbrace{A \times \ldots \times A}_{n \text{ times}}.
$$
So $(x_1, \ldots, x_n) \in A^n$ means that each $x_i$, for $1 \leq i \leq n$, is an element of $A$. Thus this is a tuple of $n$ elements of $A$ (allowing duplicate... | {
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"question_score": "2",
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Why does this cross ratio equal infinity? I'm currently studying linear fractional transformations and cross ratios and came across this in a book (this is translated from Korean, so I apologize if there are any errors or ambiguities):
We can define the cross ratio for complex numbers as:
$$[ z, z_2, z_3, z_4 ] = \dfr... | $z_4 - z_4 = 0$, so you have $0$ at a denominator. Now recall that on the Riemann sphere we have:
$$ \frac{1}{0}:= \infty $$
(note that we are not using a sign). This explains why $S(z_4)= \infty$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3686558",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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The family $f_n=\arctan(nx)$ is not equicontinous. I am studying Equicontinuous families and the book says that $\arctan(nx)$ is not equicontinuous since the definition is violated if $x=0$ I would really appreciate if someone explain me what part of the definition is violated.
My definition is: Let $F$ be a collection... | Denote $f_n(x) = \arctan(nx)$. We know that for $x>0$, $\lim_{n\rightarrow \infty} f_n(x) = \pi/2$; for $x<0$, $\lim_{n\rightarrow \infty} f_n(x) = -\pi/2$. Also, $f_n(0) = 0$. This implies $\exists \epsilon = 1 (<\pi/2)$, $\forall \delta>0$, $\exists N$ and $x\in (-\delta,\delta)$, such that when $n>N$ we have $d(f_n(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3686730",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Evaluate the determinant $\begin{vmatrix} y+z&z&y\\z&z+x&x\\y&x&x+y\end {vmatrix}$ Performing the operation $R_1\rightarrow R_1-R_2-R_3$
$$\begin{vmatrix} 0&-2x&-2x \\ y&z+x&x \\ z & x&x+y \end{vmatrix}$$
Pulling $-2x$ out and performing $C_2\rightarrow C_2-C_3$
$$-2x\begin{vmatrix} 0&0&1\\ y&z&x \\ z&-y&x+y \end{vmatr... | You made an error right off the bat: $y$ and $z$ in the first column have somehow gotten swapped after your first manipulation.
However, instead of manipulating the matrix as is, it’s often fruitful and less work to examine special cases first. Set $x=0$ and observe that the rows of the resulting matrix are linearly ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to solve the characteristic equations $\frac{dx_1}{x_1} = \frac{dx_2}{x_2} = \frac{dV}{2V}$? How to solve the following $$\frac{dx_1}{x_1} = \frac{dx_2}{x_2} = \frac{dV}{2V},$$
where $V = V(x_1,x_2)$.
My effort is that I take $$\int dx_1/x_1 = \int dV/2V + C \, \Rightarrow \, V = C_1(x_2) x_1^2$$
and
$$\int dx_2/x... | Probably you want to solve the PDE :
$$x_1\frac{\partial V}{\partial x_1}+x_2\frac{\partial V}{\partial x_2}=2V,$$
and you correctly wrote the Charpit-Lagrange system of characteristic ODEs :
$$\frac{dx_1}{x_1} = \frac{dx_2}{x_2} = \frac{dV}{2V}.$$
A first characteristic equation comes from $\frac{dx_1}{x_1} = \frac{dV... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How were complex geometric shapes drawn without computers? How did mathematicians create drawings of complex geometric shapes in the past, without 3d graphics in computers? Here is one example of what I’m talking about, drawn in the 16th century:
| Quite a bit of calculation! And very precise measurements. When I was a kid, in place of internet and all that, a person could do the calculations to draw 2D projections (or, alternatively, in-perspective versions) of aesthetically pleasing objects, such as polyhedra. Some challenging trigonometry, yes.
In those days, ... | {
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"timestamp": "2023-03-29T00:00:00",
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Suppose $f(x) \rightarrow M$ as $x \rightarrow a$. Prove that if $f(x) \leq L$ for all $x$ near $a$, then $M \leq L$. I'm am a student taking a real analysis paper at university. I'm going through some problems on my problem sheet and I've been asked the question above.
I'm still getting a hang on what it means for $x$... | Given: $\lim_{x \rightarrow a} f(x)=M$, and in a $\delta$ neighbourhood of $a$, i.e. there is a $\delta >0$ s.t.
$|x-a| \lt \delta$ implies $f(x)\le L.$
Assume $M >L$.
For $|x-a|<\delta$ we have $f(x)\le L <M.$
Let $n_0 >1/\delta$ (Archimedean principle).
For $n\ge n_0$
$|x_n-a| <1/n \le 1/n_0 <\delta$, i.e. the $x_n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3687722",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Does Thomae's function have an antiderivative? Let $f(x)=0$ if $x$ is irrational and $f(x)=1/q$ if $x=p/q\in \mathbb{Q}$, $p,q$ coprime natural numbers (restrict to $[1,2]$ to make things easier). By boundedness and as it has at most countably many discontinuous points we know that $f$ is integrable on $[1,2]$. Does it... | Here is another argument. Suppose there is an anti-derivative $F$. By Fundamental Theorem of Calculus we have $$F(x) =F(1)+\int_{1}^{x}f(t)\,dt$$ for all $x\in[1,2]$. Since $F'=f$ it follows from above equation that we have $g'=f$ where $g$ is defined by $$g(x) =\int_{1}^{x}f(t)\,dt$$ But notice that integral of Thomae... | {
"language": "en",
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How do you show that 2 lines $L_{1}$ and $L_{2}$ are perpendicular(orthogonals) but using this result $ ||u+v||=||u-v||$ My question has 2 parts, I m struggling with the second one.
Let's see the first:Prove that the vectors $u$ and $v$ are orthogonal, if and only
if $\| u + v \| = \| u - v \|$
<=
If $u,v$ are orthog... | $$\|(1,-1,1)+(5,2,-3)\|=\sqrt{41}=\|(1,-1,1)-(5,2,-3)\|.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3688121",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Residue and removable singularity I have the function $\displaystyle f(z)=\frac{z^2+z+1}{z^2(z-1)}$.
I have to calculate residue in isolated singularities (including infinity).
I calculated residue in $z = 0$ and $z = 1$, but I don't know how to calculate it in infinity. I don't understand if infinity is removable si... | The Residue at Infinity of $f(z)$ is given by
$$\text{Res}\left(f(z),z=\infty\right)=\text{Res}\left(-\frac1{z^2}f\left(\frac1z\right),z=0\right)$$
So, for $f(z)=\frac{z^2+z+1}{z^2(z-1)}$, we have
$$\begin{align}
\text{Res}\left(f(z),z=\infty\right)&=\text{Res}\left(-\frac1{z^2}\frac{1/z^2+1/z+1}{(1/z^2)(1/z-1)},z=0\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3688235",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Am I going correctly? Let $f(x)$ be a polynomial in $x$ and let $a, b$ be two real numbers where $a \neq b$
Show that if $f(x)$ is divided by $(x-a)(x-b)$ then the remainder is
$\frac{(x-a) f(b)-(x-b) f(a)}{b-a}$
MY APPROACH:-
Let $Q(x)$ be quotient so that :
$(x-a)(x-b)Q(x)+{ Remainder }=f(x)$
L.H.S, $(x-a)(x-b) \cdot... | I think you have the right idea but have not set out your argument very clearly; it looks as if you assume your conclusion about half way through. A better lay out might be:
When dividing $f(x)$ by $(x-a)(x-b)$ you obtain,
$$ f(x) = (x-a)(x-b) q(x) + r(x) $$ where $r(x)$ is a polynomial of degree one or less. Thus $r... | {
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Proving the following matrix is diagonalizable I'm asked to prove that the matrix $A\in M_{n}(\mathbb C)$ that satisfy $A^8+A^2=I$ is diagonalizable. I've tried looking at the equation $x^8+x^2-1=0$ and determining whether $M_A$ has any repeating roots, but this got me nowhere. Afterwards, I thought about trying to det... | Look at $g(x)=x^4+x-1$ . Observe that $g'(x)=4x^3+1$ has only one real root and deduce that g as no repeated root in $\mathbb R$.
Moreover, we have $g(0)=-1$ and $g(x)\to \infty$ for $x\to \infty$ and $x\to -\infty$.
So $g$ has only two distinct real roots. The other two roots are distinct and complex occurring in conj... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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A Double Integral For Calculus I I stumbled upon this questions and really messed around with it for more than half an hour but did not get any where. I checked back and forth and do not think I copied the question incorrectly.
Question: Evaluate the double integral: $\displaystyle \int_{1}^{2} \int_{\frac{1}{y}}... | Wouldn't this just be $\int_1^2\int_{1/y}^y(x^2+y^2)dxdy=\int_1^2\frac{y^3}{3}+y^3-(\frac{3}{y^3}+y) dy$?
In doing so we're treating the inner integral as simplifying the area in a sense to something of a single variable. If I did the algebra correctly after full evaluation the integral should equal $\frac{263}{24}$.
| {
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"url": "https://math.stackexchange.com/questions/3688841",
"timestamp": "2023-03-29T00:00:00",
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Stuck on volume of a solid of revolution The problem I've been working on is where the curves are y=sqrt(x), y=1/x, and x=5, which are rotated around the y-axis. I am able to do this problem when rotating around the x-axis, but I have no clue how to calculate the inner radius when neither curve allows for me to determi... | I find it often helps to make a graph of the equations, even when I think I know what the inner and outer radius will be.
For the three equations $y= \sqrt x,$ $y = 1/x,$ $x = 5$ I get the region that is shaded in the figure below.
Now your problem, if you want to use the disk/washer method, is that the curves $y=\sqr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3688974",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Closed ball is weakly closed The problem is in a Banach space, if $||x_n||\leq 1$ and $x_n\to x$ weakly, then $||x||\leq 1$
This question has an answer here: math.stackexchange.com/questions/714049/closed-unit-ball-in-a-banach-space-is-closed-in-the-weak-topology
But for convenience, I will repost the answer.
"If $x_n ... | Weak convergence does not guarantee existence of $\lim \|x\|$. So we cannot take limit on both sides of the inequality $|\lambda x_n | \leq \|\lambda\| \|x_n\|$. But we can always take $\lim \inf $ or $\lim \sup$. When you take $\lim \inf$ on both sides LHS becomes $|\lambda x|$ because if a sequence is convergent t... | {
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Why am I getting the wrong answer when I factor an $i$ out of the integrand?
Consider the following definite integral:
$$I=\int^{0}_{-1}x\sqrt{-x}dx \tag{1}$$
With the substitution $x=-u$, I got $I=-\frac{2}{5}$ (which seems correct).
But I then tried a different method by first taking out $\sqrt{-1}=i$ from the in... | If $x\in[-1,\,0)$ then $\Im\sqrt{x}=\sqrt{-x}$, so $\sqrt{-x}=\sqrt{x}/i=-i\sqrt{x}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3689229",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
"answer_count": 3,
"answer_id": 0
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Probability of an exam, what do you do? Suppose you have an exam that consists of 10 questions, each having two options to choose from. Every correctly answered question gives you 1 point, every unanswered question gives you 0 points, and every wrongly answered question gives you -0.5 points. If you want to maximise yo... | We can just consider maximizing the score on the two question out of the ten. If I don't answer, I'll get 0 points. If I answer randomly you can say I'll get 1 point from one and -0.5 from the other; so I'll get 0.5 So I should try answering.
You can repeat the same argument if you just have a question to answer.
If th... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Rudin's POMA Chapter 1 exercise 5 Hi I am writing to check if the proof that I wrote is valid, I feel like it seems kind of right to me but as I am only a beginner in writing proofs I feel like I might've missed something.
Question: Let E be a nonempty set of real numbers which is bounded below. Let -A be the set of al... | If $A$ is bounded below then there exists $M$ such that $a\in A\implies M\le a$ for all $a\in A$. Therefore, $-a\le -M$ for all $-a\in -A$ and so, by the least upper bound property of the real numbers, there exists $U$ such that $U$ is a least upper bound for $-A$.
Since $-x\le U$ for all $-x\in -A$, $-U\le x$ for all ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3689617",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Approximation of smooth diffeomorphisms by polynomial diffeomorphisms? Is it possible to (locally) approximate an arbitrary smooth diffeomorphism by a polynomial diffeomorphism?
More precisely: Let $f:\mathbb{R}^d\rightarrow\mathbb{R}^d$ be a smooth diffeomorphism. For $U\subset\mathbb{R}^d$ bounded and open and $\var... | As pointed out by Robert Bryant over at
https://mathoverflow.net/questions/364099/approximation-of-smooth-diffeomorphisms-by-polynomial-diffeomorphisms ,
the answer to this question is 'no'.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3689873",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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How do I show that if $p_n q_{n-1} - p_{n-1}q_n = 1$, then $p_n/q_n$ converges? Let $p_{n}$ and $q_{n}$ be strictly increasing, integer valued, sequences. Show that if
$$p_{n}q_{n-1}-p_{n-1}q_{n}=1,$$
for each integer $n \ge 1$, then the sequence of quotients $\frac{p_{n}}{q_{n}}$ converges.
My attempt: For given $\eps... | The question is tagged proof-writing, hence this answer is intended to address the style of the proof given, rather than to offer a more slick or elegant proof. My general critiques are as follows:
*
*The introduction of $\varepsilon$ is somewhat confusing. The argument presented relies on the fact that the sequen... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3690006",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Why does $AB + BC + CA = (A\oplus B)C + AB$ not imply $BC + CA = (A\oplus B)C$ in boolean algebra? I am new to Logical Inequalities. Please bear with me if I am inexplicably stupid.
The following is a Proven Equality:
$$AB + BC + CA = (A\oplus B)C + AB$$
I noticed that I cannot "cancel out" $AB$ from both sides of the ... | It is worth clarifying here: there is only one single rule at all times for equations: If $x=y$, then $f(x) = f(y)$, where $f$ is some mapping. Nothing more, nothing less.
What about cancellation? Isn't it true that, if $f(x) = x+ 3$, then $f(x)=f(y)$ implies $x = y$? Does this make a rule?
No. Forget it. What really h... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3690142",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 3
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An integration inequality
$f$ is differentiable on [-1,1], $M=\sup|f'|$. There is $a \in (0,1)$ such that $\int_{-a}^a f(x)dx=0$. Prove that
$$
\left|\int^1_{-1}f(x)dx\right| \le M(1-a^2)
$$
I don't know how to use $\int^a_{-a} f(x) dx = 0$.
| $$
\left| \int_{-1}^1 f(x) \mathrm{d}x \right|=
\left|\int_{-1}^{-a} f(x) \mathrm{d}x + \int_{a}^1 f(x) \mathrm{d}x\right| \\
\le \int_{-1}^{-a} |f(x)| \, \mathrm{d}x + \int_a^1 |f(x)| \, \mathrm{d}x.
$$
As mentioned in the comments, $\int_{-a}^a f(x)\mathrm{d}x=0$ implies that $f(x)=0$ for some $x\in(-a,a)$, since if... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Another series involving $\log (3)$ I will show that
$$\sum_{n = 0}^\infty \left (\frac{1}{6n + 1} + \frac{1}{6n + 3} + \frac{1}{6n + 5} - \frac{1}{2n + 1} \right ) = \frac{1}{2} \log (3).$$
My question is can this result be shown more simply then the approach given below? Perhaps using Riemann sums?
Denote the serie... | Simpler, I do not know (except with integration as @J.G. answered).
Otherwise, in order to have have a good approximation of the partial sums, you can start with
$$s_p=\sum_{n=0}^p \frac 1{a n+b}=\frac{\psi \left(p+1+\frac{b}{a}\right)-\psi
\left(\frac{b}{a}\right)}{a}$$
and consider that, for large values of $p$,
... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to show that $\{(x, y) \in \mathbb{R}^2\mid\cos(x^2) + x^3 - 4 7y > e^x - y^2\}$ is open or closed? Show that the set of points $(x, y) \in \mathbb{R}^2$ such that $\cos(x^2) + x^3 - 4 7y > e^x - y^2$ is an open subset of $\mathbb{R}^2$.
I was reading the fact that if $(X,d)$ and $(Y,d)$ are metric spaces and $f... | $f(x,y)=\cos (x^{2})+x^{3}-47y-e^{x}+y^{2}$ defines a continuous function and the given set is $f^{-1} (0,\infty)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3690568",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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A set has measure zero iff for every $\epsilon>0$ there is a countable covering of open rectangles such that $ \sum_{i=1}^\infty v(Q_i)<\epsilon $ What shown below is a reference from "Analysis on manifolds" by James R. Munkres.
Definition
Let $A$ a subset of $\Bbb{R}^n$. We say $A$ has measure zero in $\Bbb{R}^n$ if ... | Consider a rectangle $R=[a_1,b_1]\times [a_2,b_2] \times ... \times [a_n,b_n]$. For $\epsilon >0$ sufficiently small $R'=[a_1-\epsilon ,b_1+\epsilon ]\times [a_2-\epsilon ,b_2+\epsilon ] \times ... \times [a_n-\epsilon ,b_n+\epsilon ]$ contains $R$ in its interior and its volume tends to volume of $R$ as $ \epsilon \t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Recurrence relation $a_{n+1}=a_n^2-2$ Sequence $a_n$ is defined $a_{n+1}=a_n^2-2, a_0=\alpha$.
I know that a closed form for $a_n$ is $a_n=\beta^{2^n}+\frac{1}{\beta^{2^n}}$, where $\beta$ satisfies $\beta+\frac{1}{\beta}=\alpha$ and I can prove it by induction.
But this way, we have to know answer. Is there any othe... | So far as I know, there is no general method for solving non-linear difference equations, even quadratic ones like this. The equation may suggest the clever transformation $$
\begin{align}
a_n&=b_n+\frac1{b_n}\\
a_n^2&=b_n^2+\frac1{b_n^2}+2\\
a_{n+1}&=b_n^2+\frac1{b_n^2}\\
b_{n+1}&=b_n^2
\end{align}$$
but I don't know... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proof gone wrong: why don't all domains have characteristic $0$? I attempted a proof for: the characteristic of a subdomain of an integral domain $D$ is equal to the characteristic of $D$
Proof: Assume $D$ is an integral domain with characteristic $r$. Since $D$ is a ring with unity $1 \neq 0$ and no $0$ divisors, it i... | Answer to get this out of the list of unanswered questions, and to help future readers
As has been indicated in the comments, it is not always true that $n\cdot 1\neq 0$ for all $n$. For example, in a prime-order field, which is clearly a domain, $p\cdot 1=0$ and this is no contradiction.
Probably part of the misconc... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why are the $p$-oldforms $f(z)$ and $f(pz)$ linearly independent at level $\Gamma_0(pN)$? Let $f$ be a newform (normalized eigenform) of weight $k$ and level $\Gamma_0(N)$. Fix $p$ not dividing $N$ and set $f_p(z)=f(pz)$.
Viewing $f$ and $f_p$ at level $\Gamma_0(pN)$, why are they linearly independent?
Here is what... | From the Fourier series it is obvious that $f \ne cf(pz)$.
$$f(z)=\sum_n a(n) e^{2i\pi n z},\qquad f(pz)=\sum_n a(n) e^{2i\pi np z}=\sum_n a(\frac{n}p)1_{p|n} e^{2i\pi n z}$$
The sequence $a(n)$ is not a scalar multiple of $a(\frac{n}p)1_{p|n}$.
The Fourier series is uniquely defined by the function : $$a(n)=\int_{iy}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3691359",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Calculating $E[Z(Z-1)(Z-2)(Z-3)]$ where $Z$ is Poisson If $Z$ is a variable that distributes by Poisson, with Expected value, $E(Z) = 2.5$.
I need to solve:
$E[Z(Z-1)(Z-2)(Z-3)]$
So What I thought to do is first:
$E[(Z^2-Z)(Z^2-2Z)(Z^2-3Z)]$
$[E(Z^2)-E(Z)][E(Z^2)-2E(Z)][E(Z^2)-3E(Z)]$
From here I'm not sure how can I ... | Here is a plain solution, because the exercise was rather designed to have this solution path. We have:
$$
\begin{aligned}
&\Bbb E[\ Z(Z-1)(Z-2)(Z-3)\ ]\\
&=
\sum_{k\ge 0}k(k-1)(k-2)(k-3)\cdot e^{-\lambda}\frac {\lambda^k}{k!}\\
&=
\sum_{k\ge 4}k(k-1)(k-2)(k-3)\cdot e^{-\lambda}\frac {\lambda^k}{k!}\\
&=
\lambda^4\sum_... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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No sequence of rv's such that $X_n\overset{P}{\rightarrow}0$ and $\mathbb{E}(X_n)\to 2$ and also $\sup\mathbb{E}(X_n^2)<\infty$.
Prove that there is no sequence $(X_n)$ such that $X_n\overset{P}{\rightarrow}0$ and
$\mathbb{E}(X_n)\to 2$ and also $\sup_n\mathbb{E}(X_n^2)<\infty.$
Attempt. If we didn't have the last... | Assume that $X_n \to 0$ in probability and $\sup_n \mathbb{E}(X_n^2) < \infty$. Then $\mathbb{E}(X_n) \to 0$.
Indeed, let $\varepsilon >0$ and write
$$
\begin{align}|\mathbb{E}(X_n)| &\leq \mathbb{E}\left(|X_n| \mathbf{1}_{|X_n| > \varepsilon}\right) + \mathbb{E}\left(|X_n| \mathbf{1}_{|X_n|\leq \varepsilon}\right)\\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3691622",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How do I prove the floor identity $⌊x + n⌋ = ⌊x⌋ + n$ in a more precise way? I am having trouble understanding the proof provided by the author for the property stated after "Goal:".
Except from the text here is a list of useful properties:
(PROPERTY 1a) $⌊x⌋ = n$ if and only if $n ≤ x < n + 1$
(1b) $⌈x⌉ = n$ if and ... | First, the statement that $m$ is positive appears to simply be an error in the text and should be ignored (just remove the word "positive" from the proof).
Then, when it says "Suppose that $\lfloor x\rfloor =m$", that is not an assumption but rather a definition of the variable $m$. In other words, they are just defin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3691757",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Numerical approximation of derivaties I am taking a course on deep learning and I a have a good understanding of High School math.
In the course they gave a formula saying that the derivative of a function $f(x)$
$f'(x) = \frac\lim{h\to0}\frac {f(x+h)-f(x)}{h} $ (formula 1)
Which is the first principle of derivatives
... | In the first case, the error term (using Taylor) is of the form
${ 1 \over 2} h f''(\xi)$ whereas for the second it is
${1\over 4} h (f''(\xi_+)- f''(\xi_-))$ and if $f$ is smooth then
$|f''(\xi_+)- f''(\xi_-)| \le L|\xi_+- \xi_-| \le L |h|$, which is where the $h^2$ comes from.
Roughly speaking the even terms tend t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How do I calculate the limit of $\lim_{n\to \infty} (1-\frac{\theta^2}{2n^2})^{2(n+1)}$ I'm going through some physics problems about polarizers and one problem is about the case where $n+1$ polarizers are stacked up and I have to look at the case where $n \to \infty$.
Now I came up for a solution for the intensity in ... | $$\lim\left(1-\frac{\theta^2}{2n^2}\right)^{2(n+1)}=\lim\left(1-\frac{\theta^2}{2n^2}\right)^{2n^2/n}\left(1-\frac{\theta^2}{2n^2}\right)^2=\left(e^{-\theta^2}\right)^{\lim 1/n}1^2=1.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3692119",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How would you simplify the following boolean expression $(!A B)+(B !C)+(BC)+(A !B !C)$? How would you simplify the following boolean expression $(!A B)+(B !C)+(BC)+(A !B !C)$?
I factorised B and managed to get $B(!A+!C+C)+(A !B !C) = B+(A !B !C)$, but I do not know how to continue.
Using a K-map, I managed to get the r... | Actually, there is a non-intuitive (doesn't hold in ordinary algebra) Boolean algebra law that can be applied here: distributivity of disjunction over conjunction expressed as follows
$$x+yz=(x+y)(x+z).$$
Using your notation $x=B$, $y=!B$, $z=A!C$. Hence we have
$$B+!BA!C=(B+!B)(B+A!C)=1\cdot(B+A!C)=B+A!C.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3692255",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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An interesting property of a particular set of triples - multiplying two and adding the other always gives 1
Find all triples of real numbers such that multiplying any two in a
triple and adding the third always gives $1$.
When will this be the case? How can we find all such triples?
So far, I've let the numbers be... | If $ab+c=ac+b=bc+a=1$, then $a(b-c)+(c-b)=0$. Factoring out the common factor $b-c$, we get $(a-1)(b-c)=0$. Hence, either $a=1$ or $b=c$. Likewise, either $b=1$ or $a=c$, and either $c=1$ or $a=b$.
Suppose that $a=1$. Then, $b+c=bc+1=1$, so $bc=0$. Hence, one of $b$ and $c$ must be $0$ and the other must be $1$, giving... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3692438",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Matrices how to prove $A^{-1} = A$ Apologies mix up from earlier the wrong values where placed in $x_2$ and $x_3$.
Question 1
Proof that the following is true for matrix $A$, $A^{-1}$ = $A^{T}$ = $A$
$A$=
$$
1/7 \begin{pmatrix}
2 & 3 & 6 \\
3 & -6 & 2 \\
6 & 2 & -3 \\
\end{pmatrix}
$$
$A^T$=
$$
... | To check that $A^{-1}=A$, you don't need to "calculate" $A^{-1}$. If $A^{-1}=A$, then $A^2=A^{-1}A=I$; and viceversa, if $A^2=I$, then you know that $A^{-1}=A$. Here you can calculate directly that $A^2=I$.
Now, in light of the above, your calculation of $A^{-1}$ is wrong. You don't say what computations you made, so ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3692555",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How should I calculate $||\underline{u}-\underline{w}||_{2}$? I'm trying to calculate $||\underline{u}-\underline{w}||_{2}$ where:
$$
u=\begin{bmatrix}1 & 3\\
2 & 2\\
3 & 1
\end{bmatrix},\,\,\, w=\begin{bmatrix}3 & 1\\
2 & 2\\
1 & 3
\end{bmatrix}
$$
I'm not familiar with the $|| \cdot ||_2$ operator and I'm not sure ho... | What you're looking for is Matrix Norm $\|\cdot\|_p$ given by $$\|A\|_p:=\max_{|x|_p=1}|Ax|_p$$ where $|x|_p$ is the vector norm. In particular for $p=2,$ $$\|A\|_2=\sqrt{\lambda_{\max}(A^*A)}$$ where $\lambda_{\max}(A)$ denotes the largest eigenvalue of $A.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3692749",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Problem for showing a $k$-coloration of the graph $G$ Let $G$ be a $k$-chromatic graph and $$f: V (G) → [k]$$ a $k$-coloration of $G$.
Show that for each $i ∈ [1 ,. . . , k]$, there exists u ∈ $f^-1$[i] such that for each $j ∈ [1 ,. . . , k] / [i]$, there exists $v$ ∈ $N (u)$ of color $j$.
I have been quite a while stu... | here is an outline of the proof, hopefully you should be able to finalise it.
*
*Reason by contradiction.
*Start with a valid $k$-coloring, not verifying your property.
*You have one color $k$, such that for every vertex $u$ colored $k$, their neighbours don't use one color $j_u$.
*Can you change the $k$-coloring... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3693077",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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In a triangle, G is the centroid of triangle ADC. AE is perpendicular to FC. BD = DC and AC = 12. Find AB. G is the centroid of the triangle ADC. AE is perpendicular to FC. BD = DC
and AC = 12. Find AB.
According to the solution manual, we can let the midpoint of AC be H. D, G, and H are collinear as G is the centroid... |
Given AG $\perp$ CG, the midpoint H is the circumcenter of AGC, which yields GH = $\frac12$AC = 6 and in turn DH = 3GH = 18 due to the centroid point G. Then, AB = 2DH = 36 since D and H are the midpoints.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3693233",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show that $|\sin(0.1) - 0.1| \leq 0.001$ with the lagrange remainder
Show that $|\sin(0.1) - 0.1| \leq 0.001$
I know that's a basic exercise on taylor polynomial but I have made a mistake somewhere that I don't find out. Anyway, here's my attempt :
Because the function $f: \mathbb{R} \rightarrow \mathbb{R}$, $x \righ... | You also know that $\vert \sin x \vert \le \vert x \vert$. Hence
$$\vert \sin(c) \vert \frac{0.01}{2} \le 0.1 \frac{0.01}{2} = 0.0005 < 0.001$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3693584",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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How does Grinberg's theorem work? Grinberg's theorem is a condition used to prove the existence of an Hamilton cycle on a planar graph. It is formulated in this way:
Let $G$ be a finite planar graph with a Hamiltonian cycle $C$, with a fixed planar embedding. Denote by $ƒ_k$ and $g_k$ the number of $k$-gonal faces o... | Of course, before we find a Hamiltonian cycle or even know if one exists, we cannot say which faces are inside faces or outside faces. However, if there is a Hamiltonian cycle, then there is some, unknown to us, partition for which the sum equals $0$.
So the general idea for using the theorem is this: if we prove that ... | {
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"source": "stackexchange",
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Units of $R[X]/(aX-1)$ In the most upvoted answer to another question here, the author states that:
$R\to R[x]$ followed by the quotient map $R[x]\to R[x]/(ax-1)$. Call this $f$. Note that $f(a)$ is a unit in $R[x]/(ax-1)$
If I understand correctly, the result is $f(a)=ax^0\mod{ax-1}=ax^0$, and I fail to see why this... | Let the ideal $(ax-1) = I$.
The map $f$ sends $a \mapsto a+I$ in the quotient ring and $(x+I)(a+I) = ax+I$ but $-ax+1 \in I$ so $(x+I)(a+I) = 1+I$.
Hence $f(a)$ is a unit in $R[x]/I$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3694169",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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} |
Doubt in area of an infinitesimally thin ring I want to find out area of a ring of an infinitesimal width in a derivation of electrostatics. So here's how my teacher explained it. Let the inner radius of thin ring be $r$ and outer be $r + dr$.
Area of thin ring
$$dA = π(r + dr)² - πr²
= π(r² + (dr)² + 2r dr) - ... | Notice, $dr$ is infinitesimal small which tends to zero i.e. $dr\to 0$ but $dr\ne0$. Now the square of infinitesimal small length i.e. $(dr)^2$ is even much much smaller.
We can say that $2\pi rdr$ is much larger than $(dr)^2$. Thus adding $(dr)^2$ to $2\pi rdr$ doesn't make any valuable difference. Therefore $(dr)^2$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3694381",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Geometry behind $\int_{0}^{2π}\frac{e^{ix}}{e^{ix}-z}~dx=2\pi(|z|<1)$ It's a nice exercise to prove,
$$\int_{0}^{2π}\frac{e^{ix}}{e^{ix}-z}~dx=2\pi(|z|<1)$$ using Leibneiz's rule.But,what's the geometrical interpretation of this?Any idea?
| Multiplying by $i$, we get, with $w=e^{ix}$,
$$
\oint_{|w|=1}\frac{\mathrm{d}w}{w-z}=2\pi i\,[|z|\lt1]\tag1
$$
The function $\frac1{w-z}$ has residue $1$ at $w=z$, and this simply states that $z$ is inside the contour $|w|=1$ when $|z|\lt1$ and outside when $|z|\gt1$.
When $|z|=1$, $(1)$ only converges in the principal... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3694502",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
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Proving a quadratic inequality Given that $(x+y)^2 \geq 4xy$, then prove $1/x^2 + 1/y^2 \geq 4/(x^2 + y^2)$.
I have tried taking the reciprocal, squaring, square rooting, expanding, factoring and various other algebraic manipulation but nothing has worked so far.
The question seems to be structured in a way which sugg... | We can not make it because it's wrong! Try $x=0$.
The condition gives $$y^2\geq0,$$ which is true, but the statement, which we need to prove is wrong
because we can not divide by $0$.
For $xy>0$ we obtain:
$$(x+y)^2\geq4xy$$ it's $$\frac{x^2+2xy+y^2}{xy}\geq4$$ or
$$\frac{x}{y}+1+\frac{y}{x}+1\geq4$$ or
$$(x+y)\left... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3694710",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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A property of functorially finite subcategory Let $A$ be a finite dimensional algebra and $\mathcal{T}$ a full subcategory of mod$A$. $\mathcal{T}$ is said to be contravariantly finite in mod$A$ if for every module $M \in mod A$, there is some $X \in \mathcal{T}$ and a morphism $f:X \rightarrow M$ such that for every $... | This essentially follows from the results of
Auslander, M.; Smalø, Sverre O., Preprojective modules over Artin algebras, J. Algebra 66, 61-122 (1980). ZBL0477.16013.
and
Auslander, M.; Smalø, Sverre O., Almost split sequences in subcategories, J. Algebra 69, 426-454 (1981). ZBL0457.16017.
but an explicit statement and ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3694877",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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How to prove that $f(x)=x+\frac{1}{x}$ is not cyclic? Let $f(x)=x+\frac{1}{x}$ and define a cyclic function as one where $f(f(...f(x)...))=x$.
How do prove that $f(x)$ is not cyclic?
What I tried was to calculate the first composition:
$f(f(x))=x+\frac{1}{x}+\frac{1}{x+\frac{1}{x}}=\frac{x^4+3x^2+1}{x^3+x}$
Intuitive... | Note that $ f(\frac{1}{x})=f(x) $, then $f$ is not injective.
If $f^n(x)=x, \ n\geq2, $ for all $x$, then $f$ is injective, which is a contradiction.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3695087",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Lottery variance The chance to win in Lottery game is $0.1$. Michael decided to buy a ticket every week until he will win or until he will buy 5 tickets.
if X is the number of weeks Michael bought a lottery ticket,
what is the variance of $X$?
So I Calculated for X=5, which means LLLLL or LLLLLW. I calculated the proba... | The issue is that $X$ does not have a geometric distribution, precisely because you have the extra condition that $X \leq 5$. e.g. A geometric distribution would have $\mathbb{P}(X = 6) > 0$, but clearly $X$ can never be 6.
So $X$ is a discrete random variable that takes values in $\{1,2,3,4,5\}$. Let $Y$ be a geometri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3695186",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Writing explicitly $(s^2-1)^2+(t^2-1)^2$ as a polynomial in $st$ and $s+t$? Consider the symmetric polynomial
$$ P(s,t)=(s^2-1)^2+(t^2-1)^2.$$
How can we write $P$ as a polynomial in the variables $st,t+s$?
The Fundamental theorem of symmetric polynomials implies this is possible, but I am having trouble doing it in pr... | $P(s,t)=s^4+t^4-2s^2-2t^2+2$.
Now, $s^2+t^2=\sigma^2-2\pi$, where $\sigma$ and $\pi$ are the sum and product of $s$ and $t$. Thus, $s^4+t^4=(s^2+t^2)^2-2s^2t^2=(\sigma^2-2\pi)^2-2\pi^2=\sigma^4-4\pi\sigma^2+2\pi^2$.
So $P(s,t)=2-2\sigma^2+4\pi+\sigma^4-4\pi\sigma^2+2\pi^2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3695376",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Let $X$ be connected and $f:X\to\mathbb{R}$ continuous s.t. each point $x\in X$ has a nbh $U$ with $f(x)=\min_{y\in U} f(y)$. Show $f$ is constant.
Let $X$ be a connected topological space and $f:X\to\mathbb{R}$ a
continuous map such that each point $x\in X$ has a neighborhood $U$
with $f(x)=\min_{y\in U} f(y)$. S... | You are very close, in fact $V$ is open because $\forall y \in V$, $\exists U \subset X$ open such that $f(y) = \text{inf}_{a \in U}(f(a))$ so then $f(a) \geqslant f(y) \geqslant f(x)$ $\forall a \in U$. Hence $U \subset V$ so $V$ is open.
Therefore by conectedness of $X$ either $V$ or $X \setminus V$ is empty.
$V$ is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3695574",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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A is diagonalizable if and only if its minimal polynomial is a product of distinct monic linear factors I have to prove that a matrix A is diagonalizable if and only if its minimal polynomial is a product of distinct monic linear factors.
I have already proved it in one direction, meaning if f it's minimal polynomial i... | Suppose A is diagonalizable with basis $\{ v_k \}$ and corresponding eigenvalues $\{ \lambda_k \}$. If $\{ \mu_k \}$ are the distinct eigenvalues, then you can check that $(A-\mu_1 I)(A-\mu_2 I)\cdots(A-\mu_n I)$ annihilates every eigenvector and, hence, must be the $0$ matrix. So the minimal polynomial $q$ divides $p(... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Functional equation with the property $P(x+1)=P(x)+2x+1$
Find all polynomials with real-valued coefficients for which the following property $$P(x+1)=P(x)+2x+1$$ holds for all $x \in \mathbb{R}.$
This seems to be a functional equation, so the initial approach would be to try some values. For $x=0$ we would have that ... | Big hint: suppose $P(0)=c$ and work towards proving $P(x)=x^2+c$
Solution:
$$P(x+1)=P(x)+2x+1$$
Well our initial assumption from the looks of it that $P(x)=x^2+c$, but we'll work towards proving it's the only solution. First of all:
$$P(x+1)-P(x)=2x+1$$
because $P$ is a polynomial, this directly means that $P(x)=ax^2+b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3695896",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Let $A,B,X$ be sets such that $A\cup B = X$ and $A \cap B = ∅$. Show that $A = X\backslash B$ and $B = X\backslash A$. I'm trying to prove this
Let $A,B,X$ be sets such that $A\cup B = X$ and $A \cap B = ∅$. Show
that:
(1) $A = X\backslash B$
and
(2) $B = X\backslash A$.
My proof is
Let $x \in A$. We know ... | This argument is a correct but is not complete. In your proof of (1) you proved only that if $x \in A$ then $x \in X\backslash B$. This is just half of proving (1).
To have a complete proof of (1) you need to also prove the reverse implication i.e. you need to prove that if $x \in X\backslash B$ then $x \in A$.
T... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3696057",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Interpretation of $\mathbb P(A|X=x)$ in two ways Let $X:(\Omega,\mathscr A) \to (\mathbb R,\mathscr{B})$ be a random variable between two measurable spaces (the latter being the Borel measurable space over $\mathbb R$). Let $x\in \mathbb R$. Let $\mathbb P$ be a probability on $(\Omega,\mathscr A)$. Assuming $\mathbb P... | Given that the set $G := \{\omega: X(\omega) = x\}$ has positive probability, by conditional probability definition, $\varphi(X)$ satisfies
$$P(A \cap G) = \int_G \varphi(X(\omega))dP = \int_{\{x\}}\varphi(y)\mu(dy) = \varphi(x)\mu{(\{x\})} = \varphi(x)P[X = x],$$
where $\mu$ is the induced probability measure on $(\ma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3696198",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If $\lim (f(x) + 1/f(x)) = 2 $ prove that $\lim_{x \to 0} f(x) =1 $
Let $f:(-a,a) \setminus \{ 0 \} \to (0 , \infty) $ and assume $\lim_{x
\to 0} \left( f(x) + \dfrac{1}{f(x) } \right) = 2$. Prove using the
definition of limit that $\lim_{x \to 0} f(x) = 1$
Attempt:
Let $L = \lim_{x \to 0} f(x) $. Let $\epsilon > 0$... | You could not assume the limit of $f$ exists, since the statement doesn’t include it.
By the definition of limit, given $\varepsilon>0,$ there exists $\delta>0$ such that
$$2-\varepsilon <f(x)+\frac 1 {f(x)}< 2+\varepsilon$$
for $x\in (0,\delta).$ Therefore
$$|\sqrt{f(x)}-\frac 1 {\sqrt{f(x)}}|<\sqrt \varepsilon.$$
He... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3696342",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 1
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Why is the stereographic projection bijective? I know this might be a very basic question, but I am just not able to wrap my head around it.
Why is the map
$$ S:\mathbb{S}^n-\{e_{n+1}\}\rightarrow \mathbb{R}^n \quad \textrm{such that } \bar{x}\mapsto (\frac{x_1}{1-x_{n+1}},...,\frac{x_n}{1-x_{n+1}})$$
a bijective func... | It's certainly not obvious (except, like many things in mathematics, in hindsight). One can show directly that this map is both injective and surjective. It's easier perhaps to simply write down the inverse function $S^{-1}\colon \Bbb R^n \to \Bbb S^n - \{e_{n+1}\}$:
$$
S^{-1}\big( (y_1,\dots,y_n) \big) = \frac1{1+y_1^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3696478",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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digit product is $20$, digit sum is $12$, find the least number This is from a math olympiad:
The product of the digits of positive integer $n$ is $20$, and the sum of the digits is $12$. What is the smallest possible value of $n$?
I started with the prime factors of $20 = 2*2*5$.
Then tried to sum them up — $2+2+5\n... | This is too long for a comment.
So, I'm writing here. This is not a solution. I think your solution is not childish at all. Since it's a competition type problem, no one should expect a super elegant solution for such a problem. I am rephrasing for your solution.
First of all, it's impossible to have such a single-dig... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Lifting submanifolds Let $\Sigma$ be a submanifold of $M$ and let $\pi\colon \widetilde{M} \rightarrow M$ be a covering map. I would like to know if it is always true that $\pi^{-1}(\Sigma)$ is a submanifold of $\widetilde{M}$.
| Let $\hat x\in\pi^{-1}(\Sigma)$, there exists an open subset $x\in \hat U$ such that the restriction $\pi_{\mid \hat U}:\hat U\rightarrow\pi(\hat U)=U$ is a diffeomorphism.
Since $\Sigma$ a submanifold, there exits a neighborhood $V$ of $x=p(\hat x)$ a submersion $f:V\rightarrow \mathbb{R}^p$ such that $V\cap \Sigma= ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3696862",
"timestamp": "2023-03-29T00:00:00",
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Is there a real valued positive function such that it and its square integrate to $1$ Does there exist a function $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $f > 0$ and
$$
\int_{-\infty}^\infty f(x) dx = \int_{-\infty}^\infty f(x)^2 dx = 1.
$$
I suspect the answer is yes. I have looked at taking $f$ to be the PD... | For $a>0$, $p>1$, we have
$$
\int _0^{\infty} \frac{1}{(a+x)^p}\,dx=\frac{a^{1-p}}{p-1}
$$Similarly, for $b>0$, we have
$$
\int _{-\infty}^0 e^{bx}\,dx = \frac{1}{b}
$$So if we let $f(x):\mathbb{R}\to\mathbb{R}$,
$$
f(x) = \begin{cases}
e^{bx}, & x <0\\
\frac{1}{(a+x)^p},& x\geq 0
\end{cases}
$$the question amounts to ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3697053",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Curious tautological pattern on"p->p" I found something that boggles me ( I'm really a beginner in symbolic logic, so maybe it's very trivial).
I was practicing with truth-tables, and I found that:
*
*"p->p" is a tautology
*"(p->p)->p" is not a tautology.
I decided to go further, and:
*"((p->p)->p)->p" is again a... | 1) For any formula $p$, $p \to p$ is a tautology.
2) For any tautology $T$, $T \to p$ is logically equivalent to $p$. (Check it out with a truth table.)
So an even amount of occurrences of $p$ will give you tautologies (by 1)); appending another $p$ will give you something that behaves like p (by 2)). And if you take t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3697197",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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What does it mean when $|z| = 2$ is the curve in a contour integral?
B) Evaluate:
$$\oint_{|z|=2} \tan{z}\,dz$$
Specifically looking at B on the image. What is meant by $|z|=2$?
| $|z| = 2$ is the circle of radius $2$ centered at the origin. Typically, when one writes an integral like
\begin{align}
\oint_{|z| = r} f(z) \, dz,
\end{align}
what is meant is that we have to consider the path $\gamma: [0,2\pi] \to \Bbb{C}$ given as $\gamma(t) := re^{it}$ (so the orientation of the path is counter clo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3697336",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Polynomials that form $1+xy+x^2 y^2$ Show that there is no polynomials $a(x), b(x) \in R[x]$ and $c(y), d(y) \in R[y]$ such that $1+xy +x^2 y^2 = a(x) c(y) + b(x) d(y) $
| Expanding on @MikeDaas's comment we have$$\begin{align}1&=a(x)c(0)+b(x)d(0),\\1+x+x^2&=a(x)c(1)+b(x)d(1),\\1-x+x^2&=a(x)c(-1)+b(x)d(-1)\\\implies x&=\frac{c(1)-c(-1)}{2}a(x)+\frac{d(1)-d(-1)}{2}b(x),\\x^2&=\frac{c(1)-2c(0)+c(-1)}{2}a(x)+\frac{d(1)-2d(0)+d(-1)}{2}b(x).\end{align}$$Since $1,\,x,\,x^2$ are all linear comb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3697490",
"timestamp": "2023-03-29T00:00:00",
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Olympiad Minimization Problem I've been struggling to find a solution to this problem that I found in the archive of my country's Olympiad questions.
I'm particularly interested in a solution that doesn't involve the use of calculus since I know that Olympiad questions do not require the knowledge of calculus to solve... | Use AM-GM by rearranging terms creatively
Hint: A good start of using AM-GM is to consider the following:
$ \frac{ 12}{ab} + K a + L b \geq 3 \sqrt[3]{ 12 K L }$, with equality when $ \frac{12}{ab} = K a = Lb$.
$ \frac{18}{a+b} + M(a+b) \geq 2 \sqrt{ 18 M }$, with equality when $\frac{18}{a+b} = M (a+b)$.
Now, pick... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3697630",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why are there multiple base cases in this strong induction? My understanding of needing a base case, in general, is that after proving the induction step, we can assert that the proposition is true for all values from the base case.
This question ($∀n ∈ Z, n≥12$) $\implies$ ( $∃x, y ∈N$ such that $n= 4x + 5y$) uses 3 ... | Your understanding of base case is incorrect. The base cases are simply those cases needed to provide a sufficient foundation for the induction step.
Here the induction step to prove that $P(k+1)$ is true relies on knowing that $P(k-3)$ is true: if there are $x,y\in\Bbb N$ such that $4x+5y=k-3$, then clearly $$4(x+1)+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3697786",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Representing a group as a quotient of a free group Consider $G=F \rtimes T$, where $F=\mathbb{Z}_3 \times \mathbb{Z}_3$ and $T=\mathbb{Z}_5$. Let $\phi : \mathbb{Z}_5 \rightarrow Aut(\mathbb{Z}_3 \times \mathbb{Z}_3)$.
It is said that any group is the quotient of a free group? How can I represent the above group $G$ as... | You have to specify $\phi$ when there are different groups that are not isomorphic between them.
In this case $Aut(\mathbb{Z}_3 \times \mathbb{Z}_3) \cong GL(2,3)$ but the order of $GL(2,3)$ is $(3^2-1)(3^2-3)=48$ so the only $\phi$ is the trivial one.
You can always represent a finite group as a quotient of a free gro... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3697933",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Ways to obtain networks from multivariate time-series I recently became aware of a bridge between (dynamical) properties of time-series and (topological) features of an associated network representation. A variety of methods exist to embed the time-series into a network (see e.g., Transforming Time Series into Complex ... | Yes, there are so-called functional networks, where you associate each time series with a node in the network. You then determine the existence or weights of edges by applying interaction measures to the respective pair of time series. The most simple such interaction measure would be the correlation coefficient, but y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3698060",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Genus $3$ curves with a couple of distinct points $P,Q$ such that $4P \sim 4Q$ Let $C$ be a smooth curve of genus $3$ over $\mathbb{C}$. Is it true that there exist $P\neq Q \in C$ such that $4P \sim 4Q$ ? ($\sim$ denotes linear equivalence)
Notice that if $C$ is hyperelliptic then this is true (just take two different... | The dimension of the subvariety $Z$ of the moduli space $M_3$ of genus 3 curves that have a pair of points $P \ne Q$ with $4P \sim 4Q$ is 5, so it is a divisor in $M_3$.
Indeed, the linear system generated by the divisors $4P$ and $4Q$ defines a morphism
$$
f \colon C \to \mathbb{P}^1
$$
which has ramification index 4... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3698189",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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What's the relationship between Banach space and inner product space I know that Banach space is a special Hilbert space. Inner product space is a special normed space. Hilbert space is a complete inner product space. Banach space is a complete normed space. I'm wondering what is the relationship between Banach space a... | Every inner product induces a norm, so an inner product space is a normed vector space. Banach spaces are complete, normed vector spaces. So, the relation between inner product spaces and Banach spaces is that they're both normed vector spaces.
However, (1) an inner product space might lack the completeness that a Ban... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3698348",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Proving that removing any vector of the linearly dependent set gives a linearly independent set Consider the matrix representing 6 linearly dependent vectors:
$$\left(\begin{array}{llllll}
1 & 0 & 0 & 0 & 0 & 1 \\
0 & 1 & 0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 & 0 & 1 \\
0 & 0 & 0 & 1 & 0 & 1 \\
0 & 0 & 0 & 0 & 1 & 1
\end{arra... | Here the fastest method I can think of.
If (for example) the first 5 columns are linearly dependent, you can find a vector in the right kernel of the form
$$\left[
\begin{array}a
a\\
b\\
c\\
d\\
e\\
0
\end{array}\right],
$$
that is, a vector that ends with $0$. In general if you leave out the $i-th$ column and the rest... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3698549",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Can we always construct a matrix using its eigenvectors? In physics, a Hermitian matrix represents an observable and can be constructed using its eigenvalues and eigenvectors in the following way:
$$ A = \sum_i \lambda_i v_iv_i^\dagger \qquad \qquad (1)$$
where $\lambda_i$ and $v_i$ are the $i^{th}$ eigenvalue and eige... | If the eigenvalues of $A$ are real, but $A\neq A^\dagger$, then the right hand side of (1) is Hermitian, but the left hand side is not, so (1) fails. An example is $$\left(\begin{array}{cc} 1&1\\0&2\end{array}\right)$$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3698916",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Proving $\operatorname{cos}(x+y)=\operatorname{cos}(x)\operatorname{cos}(y)-\operatorname{sin}(x)\operatorname{sin}(y)$ using differentiation While proving $\operatorname{cos}(x+y)=\operatorname{cos}(x)\operatorname{cos}(y)-\operatorname{sin}(x)\operatorname{sin}(y)$
by this $$\operatorname{sin}(x+y)=\operatorname{si... | To elaborate on my comment about taking a partial derivative. You are assuming that $y = y(x)$. Your expression is of the form
$$f(x,y)(1 + y'(x)) = g(x,y)(1 + y'(x))$$
and your are trying to conclude the equality
$$f(x,y) = g(x,y), \ \forall (x,y)\in \mathbb{R}^2 $$
But you choose $y$ to be an arbitrary function of $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3699074",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
$P(x)=P(-x)$ holds for all values of $x$ ,two conditions I was doing a question on polynomials , where it was found that
$P(x)=P(-x)$ in the interval $[-\sqrt2,\sqrt 2 ]$
It was then concluded that $P(x)=P(-x)$ holds for all values of $x$ ,"since it is a polynomial".
Can someone help me understand why it could be gen... | Let $Q(x)=P(x)-P(-x)$, then $Q(x)$ is a real polynomial since $P(x)$ is (make sure you can show this!). By the assumption, $Q$ has infinitely many roots. But the only real polynomial with infinitely many roots is the zero polynomial. Hence $Q(x)=0$ for all real $x$, so $P(x)=P(-x)$ for all real $x$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3699219",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Matrix of the differentiation operation Exercise: Find the matrix of the derivative operation $D$ related to the base $\{1, t, t^2,..., t^n\}$ $$D: \mathcal P_{n} \to \mathcal P_{n}$$
I found a possible solution to this exercise, given that $D(t^k)=kt^{k-1}$ $$ \begin{equation*}
D_{n+1,n+1} =
\begin{pmatrix}
0 & 1 & 0... | The numbers in your vectors represent the linear combination of the basis elements needed to form a polynomial. For example,
\begin{equation*}
v =
\begin{pmatrix}
3 \\
4 \\
\vdots \\
6 \\
7 \\
\end{pmatrix}
\end{equation*}
The polynomial represented by this vector is $3+4x+...6x^{n-1}+7x^n$.
Now, for a polynomial l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3699311",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
How to visualize Euler's number? I am interested if there is geometric meaning (using graphs) of $(1 + \frac{1}{n})^n$ when $n \rightarrow \infty$. Also, is there visual explanation of why is $e^x = (1 + \frac{x}{n})^n$ when $n \rightarrow \infty$ and why is $\frac{d}{dx}e^x = e^x$?
I see that this kind of question is ... | I think of my favorite, and pretty geometric, proof of this limit, using the squeeze or sandwich theorem for limits. You can do it using an upper and lower Riemann sum with one subdivision for the integral of $1/t$.
One has $L\le\int_1^{1+x/n}1/t\rm dt\le U\implies x/n(1/(1+x/n))\le\ln(1+x/n)\le x/n(1)\implies x/(n+x)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3699439",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Characteristic Subspace of a k-form I have to show that two decomposable k-forms on an n-dimensional vector space $\Bbb V^n$ have the same characteristic subspace (of dimension $n-k$) if and only if one is scalar multiple of the other.
$\underline{\text{Definition}}$: Characteristic subspace of a $k$-form $\omega$ on $... | Let $Z$ denote the characteristic subspace, and let $\{z_1,\cdots,z_{n-k}\}$ be a basis for $V$. You can extend this to a basis $\{z_1,...,z_{n-k},w_1,...,w_k\}$ of $V$. Define an inner product on $V$ by saying the dot product of a vector in the basis with itself is $1$, and is $0$ with any other vector in the basis, a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3699555",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Schlag's proof of the bigPicard theorem I am trying to understand the proof of Picard's big theorem which is theorem $4.20$ in Wilhelm Schlag's book A course in complex analysis and Riemann surfaces.
The theorem is stated as follows
If $f$ has an isolated essential singularity at $z_0$, then in any small neighborhoo... | He is proving that a function holomorphic in a annulus $\{z:0<|z|<R\}$ which omits
two values has either a removable singularity or a pole at $z=0$, and certainly
does not have an essential singualarity there.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3699695",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Show $n I_n I_{n-1} = \frac{\pi}{2}$ where $I_n = \displaystyle\int_0^{\pi/2} \cos^n x dx$. Consider the sequence $(I_n)_{n \in \mathbb{N}}$:
$$ I_0 = \frac{\pi}{2} $$
$$ I_n = \int_0^{\pi/2} \cos ^n x dx$$
For this sequence I have to prove that the following is true:
$$n I_n I_{n-1} = \frac{\pi}{2}$$
for $n \in \mathb... | I'll continue from where you got to :
$$nI_{n}I_{n-1} = (n-1)I_{n-2}I_{n-1} $$
Now,
$$nI_{n}I_{n-1} = (n-1)I_{n-2}I_{n-1}$$
$$(n-1)I_{n-1}I_{n-2} = (n-2)I_{n-3}I_{n-2}$$
$$(n-2)I_{n-2}I_{n-3} = (n-3)I_{n-4}I_{n-3}$$
$$.....$$
$$(2)I_2I_1 =I_0I_1$$
Multiply all these equations: You can see most of the terms get cancell... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3699899",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
$C(S)$ is dense implies $S$ is metrizable given $S$ is compact and Hausdorff For a compact Hausdorff space $S$, show that the following are equivalent:
(a) $S$ is metrizable.
(b) $S$ has a countable base.
(c) $C(S)$ is separable, where $C(S)$ denotes the space of all continuous functions on $S$.
(a)$\implies$(b) This ... | Show that $d$ is continuous on $S \times S$. Also show that $d$ is actually a metric on $S$ (there is something to be shows for $d(x,y)=0 \to x=y$, e.g.)
Let $\mathcal{T}_d$ be the topology on $S$ that is generated by $d$ (i.e. in which the balls $B_d(x,r)$ are open for all $x \in S$ and all $r>0$), and $\mathcal{T}$ t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3700164",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
What's the distribution of $xy+xz+yz$ where $x,y,z $ are independent standard normal? We know the product of two independent Normal random variables has a normal product distribution, or Variance Gamma distribution if they are correlated.
But, what if there are three Normal random variables?
So, here is the question: ... | The probability density function $f$ is given by
$$f(x)=\begin{cases}
\dfrac1{\surd3}\mathrm e^x & \text{if $x<0$}, \\
\dfrac2{\surd3}\mathrm e^x[1-\Phi(\sqrt{3x})] & \text{if $x\geqslant0$}, \end{cases}$$where $\Phi(x):=\dfrac1{\surd(2\pi)}\int_{-\infty}^x\exp\dfrac{-t^2}2\mathrm dt$ is the standard-normal cumulative ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3700360",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Reference for very basic books in Functional analysis I'm confused about which books I have to read for Functional analysis for the beginner level.
I need references for very basic books in Functional analysis and that book must contain given Topics below
$1.$ Normed linear spaces,
$2.$ Banach spaces,
$3.$ Hilbert ... | My top two recommendations would be Functional Analysis: A First Course by M.T Nair (very beginner friendly) and Introductory Functional Analysis with Applications by E. Kreyszig.
I also recommend the notes by V. S. Sunder for more of a spectral theoretic focus. You can also check out Functional Analysis by S. Kesavan... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3700489",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Definition of eigen space. I am studying linear algebra and got confused in defining eigen space corresponding to eigen value.The thing wondering me is the same thing defined in two different books in different manners.let $\lambda$ be eigen value of matrix $A$ of order n over the field $\mathbb{F}$ then Hoffman & kunz... | You are right: they are not equivalent. The first definition is the usual one. The second one is more general: if $F_\lambda$ is the eigenspace corresponding to $\lambda$ (with respect to the first definition), then the author of the second definition is saying that any non-zero subspace of $F_\lambda$ is an eigenspace... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3700655",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Prove the difference is more than $n$ and less than $2n$ We chose $n + 2$ numbers from the set $\{1,2,....3n\}$ . Prove that there are always two among the chosen numbers whose difference is more than $n$ but less than $2n$.
Though I can understand it by taking examples but I really struggle when it comes to prove. Als... | Like Alexey suggested you can consider the remainders of division by $n+1$ of these $n+2$ numbers.
Since the different possibile remainders are only $n+1$ and you have $n+2$ different possible solutions you know, thanks to the pidegeonhole principle, that at least $2$ different numbers, $x$ and $y$, have the same remai... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3700773",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Let $f:\mathbb R^n\to \mathbb R$ be a continuous function with the property that there exits two sequence $(b_n)$ and $(c_n)\in \mathbb R^n$ such that Let $f:\mathbb R^n\to \mathbb R$ be a continuous function with the property that there exits two sequences $(b_n)$ and $(c_n)\in \mathbb R^n$ such that
$f(b_n)\to \infty... | Suppose that $lim_n\|b_n\|$ is not $\infty$, there exists $C$ such that for every $n$, there exists $p_n>n$ with $\|b_n\|<C$. Let $B(0,C)$ be the closed ball of radius $f(B,0,C))$, since $f$ is continuous, $f(B(0,C))$ is compact and bounded, and contained in $[-P,P]$. Since $lim_nf(b_n)=\infty$, there exists $n_P$ s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3700940",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Calculating $\frac{1}{a}$ using Newton-Raphson method I have a computer that doesn't implement division operation (it has only addition, substraction and multiplication). I need to find a method to find the approximate value of $\frac{1}{a}$, where $a\in \mathbb R \setminus\{0\}$. I'm supposed to do that with Newton-Ra... | We take function: $f(x)=a-\frac{1}{x}$.
$$
x_{k+1}=x_k-\frac{a-\frac{1}{x_k}}{\frac{1}{x_k^2}}=\frac{\frac{1}{x_k}-a+\frac{1}{x_k}}{\frac{1}{x_k^2}}=2x_k-a\cdot x_k^2
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3701131",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Uniform convergence and integrals. I'm asked to tell if the following integral is finite:
$$\int_0^1 \left(\sum_{n=1}^{\infty}\sin\left(\frac{1}{n}\right)x^n \right)dx$$
I studied the series (which converges uniformly on $(-1,1)$ by d'Alembert's Criterion and in $-1$ by Leibniz's Criterion, so in general the convergenc... | We know that $f_{n}(x)= \sin \Big( \frac{1}{n} \Big) x^{n}$ are Lebesgue integrable on $(0,1)$ and positive. Since $$ \sum_{n=1}^{ \infty} \int_{0}^{1} \sin \Big( \frac{1}{n} \Big) x^{n} dx = \sum_{n=1}^{ \infty} \sin \Big( \frac{1}{n} \Big) \frac{1}{n+1}, $$ which converges, because $$ \sin \Big( \frac{1}{n} \Big) \le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3701461",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What is the coordinate definition of holomorphic vector fields? We can write a holomorphic vector field in local co-ordinates as $X=X^i\dfrac{\partial}{\partial z_i}$, where $\dfrac{\partial}{\partial z_i}$ forms a local frame for $T^{(1,0)}M$. My questions are
*
*Does $X^i$ have to be holomorphic functions?
*Can w... | *
*is right due to the definition of the holomorphic vector field.
*is also right.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3701628",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
$\ker(T) + \ker(S) = V \implies {\rm Im}(T + S) ={\rm Im}(T) +{\rm Im}(S)$ Let $V$ be a vector space. Let $T, S$ be two linear operators $T:V \rightarrow V$, $S: V \rightarrow V$, such that $$\ker(T) + \ker(S) = V$$, then we must have $${\rm Im}(T+S) = {\rm Im}(T) + {\rm Im}(S)$$.
If the statement is true, then give th... | True.
Proof: Clearly $\text{Im}(T+S) \subseteq \text{Im}(T)+\text{Im}(S)$, so we just need to show $\text{Im}(T)+\text{Im}(S) \subseteq \text{Im}(T+S)$. As another answer suggested, it suffices to show $\text{Im}(T) \subseteq \text{Im}(T+S)$ and $\text{Im}(S) \subseteq \text{Im}(T+S)$, because in general if $U, W$ are ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3701793",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Prove that if $E[X_1^p]<\infty$, then $\frac{\max_{1\le i\le n} X_i}{n^{1/p}} \rightarrow 0$ in probability where $\{X_n\}$ is i.i.d and non-negative Suppose $\{X_n\}_{n\geq 1}$ are iid and non negative. Define $M_n=\max \limits_{i=1,\ldots,n}\{X_i\}.$
Prove if $E[X_1^p]<\infty$, then $\frac{M_n}{n^{1/p}}\rightarrow 0... | You can use the result $P[M_n>x]\leq nP[X_1>x]$. Observe that
$$P[M_n/n^{1/p}>\epsilon]=P[M_n>\epsilon n^{1/p}]
\le nP[X>\epsilon n^{1/p}]=nP[X^p>\epsilon^pn].$$ Now use the inequality $P[Y>a]\le E\left[\frac YaI(Y>a)\right]$ to conclude $$nP[X^p>\epsilon^pn]\le E\left[\frac{X^p}{\epsilon^p}I(X^p>\epsilon^pn)\right],$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3701965",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Basic question about inequality let $x>1$
Obviously $x^2>x$
Then $x^2>x>1$
Taking $x^2>1$, we can assert that this holds true for all the values for $x>1$ and $x<-1$
But if I take $-5$ such that $x<-1$, then $x^2>x$ holds but $x>1$ doesn't. Why is it so? Doesn't $x^2>x>1$ mean that all of the three must be true?
| When you deduced that $^2>>1$, you specifically had the constraint that $x > 1$. Of course, this inequality is not then applicable since now you have $x > -5$ which is a larger domain than $x > -1$.
More specifically, $x > 1 \Rightarrow x^2 > x > 1$ is a true statement, but $x^2 > x \iff x> 1$ is clearly not a true st... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3702139",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
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