Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Category Theory: special case of commutative diagram / function I am sorry if this is an obvious question.
I have a curious situation, in a software design, where it's supposed that there may exist a space $N$ (for normalised data) and a function $e: N \to K$ ($K$ another space different from $N$), such that for each f... | Your condition means precisely that $e$ is a retraction, i.e. that $e$ admits a right inverse, i.e. that there exists a morphism $f \colon K \to N$ with $e \circ f = \operatorname{id}_K$.
To see that such an $f$ exists under the given conditions we take $L = K$ and $h = \operatorname{id}_K$ and then take for $f$ the re... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3238616",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to evaluate the following integral involving a gaussian? I want to evaluate the following integral:
$$\int\limits_0 ^\infty {x \sin{px} \exp{(-a^2x^2})} dx$$
Now I am unsure how to proceed. I know that this is an even function so I can extend the limit terms to $-\infty, \infty $ and then divide by 2. I have tried ... | Start with:
$$I\left( p \right)=\int_{0}^{\infty }{\cos \left( px \right)\exp (-{{a}^{2}}{{x}^{2}})dx}$$
We can use differentiation under the integral sign:
$${I}'\left( p \right)=-\int_{0}^{\infty }{x\sin \left( px \right)\exp (-{{a}^{2}}{{x}^{2}})dx}$$
Integration by parts using $u=\sin \left( px \right)\quad and\qua... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3238744",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
} |
If ${}^nP_{12}={}^nP_{10}×6$, than what is $n$?
If ${}^nP_{12}={}^nP_{10}×6$, than what is $n$?
I am at year 11. I do understand the concept of $^nP_r,{}^nC_r$. Once I know the $n$ I can calculate. I got stuck on this.
| So to compute nPr we do $n(n-1)\ldots(n-r+1)$. We note the following recurence
\begin{equation}
nPr = nP(r-1) \times (n-r+1)
\end{equation}
If you apply this recurrence twice, you should find a quadratic equation that $n$ must satisfy. When you find the solutions, pick the one such that both $nP12$ and $nP10$ make sens... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3238866",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
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Prove $(y-x^2)$ is a prime ideal in $\mathbb{R}[x,y]$, but not maximal. My guess is to use the fact that when we take the quotient,
$\mathbb{R}[x,y]/(y-x^2)$, this will become an integral domain but not a field.
I am not sure how to take the quotient, though. I am also unfamiliar with the ring of polynomials of two var... | Indeed, $x\mapsto x$, $y\mapsto x^2$ gives us a homomorphism $\Bbb R[x,y]\to\Bbb R[x]$. Show that the kernel is $(x^2-y)$ and you are done as that shows $\Bbb R[x,y]/(x^2-y)\cong \Bbb R[x]$, which is an integral domain, but not a field.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3238973",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Solve $\sqrt{1 + \sqrt{1-x^{2}}}\left(\sqrt{(1+x)^{3}} + \sqrt{(1-x)^{3}} \right) = 2 + \sqrt{1-x^{2}} $ Solve $$\sqrt{1 + \sqrt{1-x^{2}}}\left(\sqrt{(1+x)^{3}} + \sqrt{(1-x)^{3}} \right) = 2 + \sqrt{1-x^{2}} $$
My attempt:
Let $A = \sqrt{1+x}, B = \sqrt{1-x}$ and then by squaring the problematic equation we get:
$$(... | First, this is stated as an equation to solve (for $x$) rather than an identity to be shown.
So with $a=\sqrt {1+x}$ and $b=\sqrt {1-x}$ we have $$a^2+b^2=2$$ and $$(a+b)^2=a^2+2ab+b^2=2(1+ab)$$ and $$a^3+b^3=(a+b)(a^2-ab+b^2)=(a+b)(2-ab)$$
Then $$\sqrt {1+ab}\cdot (a^3+b^3)=\frac {\sqrt 2}2(a+b)(a+b)(2-ab)=\sqrt 2(1+a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3239071",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 7,
"answer_id": 0
} |
exponential time complexity for $M(n,n)$ with $ M(i,j) = M(i-1,j) + M(i-1,j-1) + M(i,j-1) $. For $n \in \mathbb{N}$ we define $Q(n) = M(n,n)$ with:
$$ M(i,j) = M(i-1,j) + M(i-1,j-1) + M(i,j-1) $$ and
$$ M(i,0) := M(0,i) := i \mbox{ } \mbox{ } \forall i \geq 0 $$
Show that $Q(n)$ (regarding the recursion above ) does ta... | Let the number of function calls required for computing $M(i,j)$ the naive way be $F(i,j)$. Then for sufficiently large $n$, and since $F()$ and $M()$ are symmetric in their arguments,
$$F(n,n)=F(n,n-1)+F(n-1,n)+F(n-1,n-1)=2F(n,n-1)+F(n-1,n-1)$$
$$=2(F(n,n-2)+F(n-1,n-1)+F(n-1,n-2))+F(n-1,n-1)$$
$$>2F(n-1,n-1)+F(n-1,n-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3239166",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
How can I evaluate $\lim_{x \to \infty}\frac{\sum_{k=1}^{\ 1000}(x+k)^{10}}{x^{10}+10^{10}}$? I would like to examine the limit of the following function as x goes towards infinity:
$$\lim_{x \to \infty}\frac{\sum\limits_{k=1}^{\ 1000}(x+k)^{10}}{x^{10}+10^{10}}$$
I already have tried to split the sum, exlude the fact... | Here the "honest" calculation using properties of limits:
\begin{eqnarray*} \frac{\sum_{k=1}^{\ 1000}(x+k)^{10}}{x^{10}+10^{10}}
& = & \frac{\sum_{k=1}^{\ 1000}\left(1+\frac{k}{x}\right)^{10}}{1+\frac{10^{10}}{x^{10}}}\\
& \stackrel{x\to \infty}{\longrightarrow} & \frac{\sum_{k=1}^{\ 1000}\left(1+0\right)^{10}}{1+0} \\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3239253",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Definition of infinite cyclic group I'm having some conceptual issues with the infinite cyclic group $C_\infty$. Finite groups $C_n$ have a clear representation as integers $0,1,\cdots,n-1$ under addition $\pmod{n}$, or as the rotation group of the $n$-gon for $n\geq 3$. The rotation group of a circle, which is what I ... | The (up to isomorphism) infinite cyclic group is just $\mathbb{Z}$ under addition.
You can visualize it as the group of integer shifts of the integers.
You can also visualize it as the rotations of the circle through integer numbers of radians, but that's not pretty geometrically since the orbit of any point on the cir... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3239387",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Modular exponentitation (Finding the remainder) I want to find the remainder of $8^{119}$ divided by $20$ and as for as i do is follows:
$8^2=64\equiv 4 \pmod {20} \\
8^4\equiv 16 \pmod {20} \\
8^8\equiv 16 \pmod {20}\\
8^{16}\equiv 16 \pmod {20}$
from this i see the pattern as follows $8^{4\cdot 2^{n-1}} \text{is alwa... | Although $8^{4\cdot 2^{n-1}}\equiv 16\pmod {20} $ for all positive integer $n $, it is actually much simpler:
$8^{4k}\equiv 16\pmod {20} $ for all $k>0$. That is for any multiple of $4$, not just $4$times powers of $2$.
And furthermore $8^{4k+1}\equiv 16*8\equiv -4*8\equiv-32\equiv 8\pmod {20} $ for $k>0$
And $8^{4k+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3239872",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Inverse of an upper bidiagonal Toeplitz matrix I have a matrix with the following structure
$$\left[\begin{array}{cccccc|c}
-1 & 1-b & 0 & \dots & 0 & 0 & b \\
0 & -1 & 1-b & \dots & 0 & 0 & b \\
\cdots \\
0 & 0 & 0 & \dots &-1 & 1-b & b \\
0 & 0 & 0 & \dots & 0 & -1 & 1 \\ \hline
... | Elements $m_{ij}$ of the inverse matrix $M=A^{-1}$ are:
\begin{align}
m_{ij}
&=
\begin{cases}
-(1-b)^{j-i}, &j\ge i,
\\
0, &j<i.
\end{cases}
\end{align}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3239967",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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The set of all values of m for which $mx^2 – 6mx + 5m + 1 > 0$ for all real x is The set of all values of m for which $mx^2 – 6mx + 5m + 1 > 0$ for all real x is?
The answer given is $0<=m<1/4$
My working: $D>=0$
$=> (-6m)^2 -4(m)(5m+1)>=0$
$=> m(4m-1)>=0$
=> Either $m>=1/4$ or $m<=0$
Where am I going wrong?
| Option:
$y=m(x^2-6x+5) +1>0;$
0) $m=0$√
1) $m>0$.
A parabola opening upward.
Minimum at:
$y'=m(2x-6)=0;$ $x=3;$
$y_{\min}=m(9-18+5)+1=$
$-4m+1$;
We require: $y_{\min}= -4m+1>0$, or $m<1/4$;
Combining : $0 \le m < 1/4$.
2) Rule out $m<0$
(Parabola opening downward )
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3240052",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
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LCM of irrational numbers So i read in a book that irrational and rational numbers do not have a common multiple and it said that lcm of irrational numbers is also only possible when both the irrational numbers have the same surd. I was wondering what this means.
| Those are odd claims.
The first can, I think, be justified. Let's say $\alpha$ is an irrational number and $\frac ab$ is rational (with $a,b\in \mathbb Z$). Then it is certainly true that, for any non-zero integers $m,n$ we have $m\times \alpha$ is irrational and $n\times \frac ab$ is rational, so it is not possible ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is there a general effective method to solve Smullyan style Knights and Knaves problems? Is the truth table method the most appropriate one? Below, an attempt at solving a knight/knave puzzle using the truth table method.
Are there other methods?
Source : https://en.wikipedia.org/wiki/Knights_and_Knaves
| Truth tables always work, of course, but another appproach is to use algebra in the field with two elements (i.e. the integers modulo 2) to represent truth values.
If we let $1$ represent a true statement and $0$ represent a false statement, then
*
*"X and Y" corresponds to $xy$.
*"not X" corresponds to $1-x$.
*"X... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3240256",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 1
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Prove that if $p\mid ab$ where $a$ and $b$ are positive integers and $a\lt p$ then $p\le b$ I have found an old textbook called "Real Variables by Claude W. Burrill and John R. Knudsen" in the first chapter this textbook uses 15 axioms to derive much of the well known and basic facts about the integers, i have been rea... | As a way to suggest that this is at least nearly equivalent to Euclid (or something like it), let's see how it does with the so-called Hilbert Numbers. These are just the naturals of the form $4k+1$. They are useful for thinking about things like unique factorization, since such basic properties do not hold for them.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3240319",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Find the probability that no two among $A$, $B$, and $C$ are together when $12$ people are arranged in a circle There are $12$ people including A,B and C. They are arranged in a circle. Find the probability that no two among A, B and C are together.
I have solved problem where cases involving two person sitting adjacen... | I think it is easier to find the probability where the condition is not satisfied. We have two cases to consider:
Case 1: A,B,C are adjacent. This is simply grouping A,B,C and changing their places among themselves with $3!$ and merging them with the remaining $9$ people. In this case, there are
$$3!\cdot(10-1)! = 3!\c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3240435",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Tensor coordinates problem
Let $B = ((1,2)^T,(1,3)^T)$ be the basis of $V=\Bbb R^2$.
Find the dual basis $B^*=(e^1,e^2)$
Find the matrix of the billinear form (tensor): $ T =
e^1\oplus e^2 - e^2\oplus e^1 + 2e^2 \oplus e^2$ with the respect to
the canonical basis.
I suppose my tensor is given with the respect to ... | The correct symbol $\otimes$ is produced with \otimes. A strategy is to first find the matrix of $T$ taken with respect to the basis $\mathcal{B}$, and then convert it to the standard basis. With respect to $\mathcal{B}$, it is clear that $$[T]_{\mathcal{B}} = \begin{pmatrix} 0 & 1 \\-1 & 2 \end{pmatrix}.$$Now, we use ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3240535",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Divisibility by 3 Maybe this is a duplicate question (if so, can the moderators be kind enough to merge this appropriately?), but what is the condition on $a$ and $b$ for an expression $am+b$ to be divisible by $3$ ($a$ and $b$ are integers)? For example, I can say $16m+3$ is divisible by $3$ since $b=3$ is divisible... | A basic principle is that if $3|v$, then $3|w \iff 3|v+w$. Furthermore, if $3|v$ then
$3|nv$.
So $3|ma+b\iff3|ra+s,$ where $r$ is the remainder when $m$ is divided by $3,$
and $s$ is the remainder when $b$ is divided by $3$.
Now we have only $9$ possibilities to consider:
$r=0, 1, $ or $2, $ and $s=0, 1, $ or $2$.
W... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3240676",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
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Existence of continuous $r(t)$ with $\lim_{t \to \infty} \frac{f(r(t))}{g(t)} = 1$ Let $ \ f,g: \mathbb{R} \to \mathbb{R} \ $ be continuously differentiable functions such that $$\lim_{t \to \infty} f(t) = \infty = \lim_{t \to \infty} g(t) \ \ . $$ My question is:
Is there a continuous function $ \ r: \mathbb{R} \to \... | Consider the functions $f(t)=t(2+\cos t)$ and $g(t)=t$. They are infinitely differentiable at the whole domain. Now suppose that the above function $r$ exists.
Let $T>0$ be large enough for $\frac 2 3 < \frac {f(r(t))}{g(t)} < \frac 4 3$ to hold for all $t>T$. This implies $$T<t_1<t_2 \implies \frac {f(r(t_1))}{f(r(t_2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3240786",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
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Every functor $\mathcal C\to\mathsf{Set}$ is an epimorphic image of a monofunctor implies every morphism of $\mathcal C$ is monic I am a trying to solve the following problem.
A functor $F: \mathcal{C} \rightarrow \mathsf{Set}$ is called a monofunctor if $F(f)$ is a monomorphism (that is, injective) for every morphism... | Hint 1
We have some $G:\mathcal C\to\mathbf{Set}$ such that $\alpha:G\twoheadrightarrow\mathcal{C}(A,-)$ for an object $A$. This means $\alpha_B:GB\twoheadrightarrow\mathcal C(A,B)$ for all $B$. In particular, for $\alpha_A$ we have a surjection $GA\twoheadrightarrow\mathcal C(A,A)$ which means there's an element $\eta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3240939",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Solution of $ty'' +(2t+3)y' +(t+3)y = 3e^{-t}$ via Laplace transform A recent question which was put on hold due to lack of context by the OP was the following:
Solve the following ODE using Laplace transforms.
$$ty'' +(2t+3)y' +(t+3)y = 3e^{-t}, \qquad y(0)=0$$
Putting the equation into the form
$$t(y^{\prime\prime}... | Keeping in mind that
$$
\mathcal{L}(t\mathcal{D}(y)) = -\frac{d}{ds}\mathcal{L}(\mathcal{D}(y))
$$
assuming null initial conditions and applying this in
$$
t \mathcal{D}^2y + \mathcal{D}y = 3e^{-t}
$$
we have
$$
-\frac{d}{ds}\left((s+1)^2 Y\right)+(s+1)Y = \frac{3}{s+1}
$$
or
$$
-Y'+\frac{1}{s+1}Y = \frac{3}{(s+1)^3}
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3241038",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Question regarding surjective mapping I have come across a question while solving practice papers on the topic 'Functions'.
The question is as follows -
If $f : X \to Y $, find $f (X)$, when $f $ is a surjective or onto mapping.
Here $X $ and $Y $ are non-empty sets
Here is my approach -
As $f $ is a surjective mapp... | Just to be really really pedantic, let us prove it by double inclusion. Let us set $f(X):=\left\{y\in Y\mid \exists\,x\in X \text{ such that } y=f(x)\right\}$. Thus $f(X)\subseteq Y$. On the other hand, surjectivity means that for every $y\in Y$ there exists $x\in X$ such that $y=f(x)$, that is to say, that $Y\subseteq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3241183",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Help with sage math defining function of two variables Hello I need help regarding sage math, since I cant find anything about it on the manuals.
So I have a function of the form $F(r,t) = 2r H(t)$ and then i want to perform an operation on it involving differentiation. H(t) is kept arbitrary. I know how to do this u... | I think this is what you want.
var('t,r')
H = function('H', nargs=1)(t)
F = 2*r*H(t)
print F
print diff(F,r)
print diff(F,r,t)
var('u,v')
G = F.subs(r=(v-u)/2,t=(u+v)/2)
print G
print diff(G,u)
print diff(G,u,v)
output:
2*r*H(t)
2*H(t)
2*diff(H(t), t)
-(u - v)*H(1/2*u + 1/2*v)
-1/2*(u - v)*D[0](H)(1/2*u + 1/2*v) - H(1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3241273",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Properties of a derivative on a compact interval Suppose a function $F$ is differentiable on an interval $(a,b) \supset [0,1]$. Denote its derivative by $f$, and suppose that $f > 0$ on $[0,1]$.
Question 1: Is it true that $f$ can be bounded away from $0$ on $[0,1]$, i.e. that there exists some $c > 0$ such that $f(x) ... | No. For instance, let $f$ be piecewise linear and positive on $[0,1)$ such that on an infinite sequence of intervals approaching $1$, $f$ alternates between jumping down to values approaching $0$, jumping up to values approaching $\infty$, and jumping back down to $1$ and remaining constant with value $1$. Define $F(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3241426",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Recurrence relations and power series solution I am given the following initial value problem: $$(1-x^2)y''+7xy'-26y=0 \qquad , \qquad y(0)=0 \qquad , \qquad y'(0)=4$$
I have solved for the singular points, which are $x= 1, -1$
The question then tells me that I can find a normal power series solution for $y$ about $x... | Considering the differential equation
$$
(1-x^2)y''+\alpha x y' + \beta y = 0
$$
and substituting $y = \sum_{k=0}^n a_k x^k$ we obtain the recurrences
$$
2a_2+\beta a_0 = 0\\
6a_3+(\alpha+\beta)a_1 = 0
$$
and for $k \ge 4$
$$
k(k-1)a_k +((k-2)\alpha +\beta-(k-2)(k-3))a_{k-2} = 0
$$
Attached a plot showing in black the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3241563",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Example of exotic $S_5$ as a Galois group Is there an example of a sextic irreducible polynomial over $\mathbb{Q}$ with Galois group isomorphic to $S_5$?
The transitive action of the Galois group of this polynomial on the 6 roots of $p(x)$ would give rise to the exotic embedding $S_5\to S_6$.
| The Galois groups of sextic polynomials have been determined here. In table $2$ on page $5$, the group T14 is $S_5$, generated by $(15364), (16)(24), (3465)$ in $S_6$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3241698",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Trying to show that logarithm of this matrix does not convergence let $A \in O(3) - SO(3)$, i.e, $A$ is an orthogonal $3 \times 3$ matrix with real entries that has a determinant $-1$. I'm trying to show that $logA$ defined as:
\begin{equation*}
logA = (A-I) - \frac{(A - I)^2}{2} + \frac{(A - I)^3}{3} - \frac{(A - I)^... | Let's break this down. If a matrix $A$ has determinant $d$, then $\log A$ will have determinant $\log d$. Can we get a sum with determinant $\log d$ from a series of real matrices when $d=-1$?
Yup, it's bad.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3241790",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A function $g(x)$ has one and only one real root if $g'(x)\leq k <0$. $g : \mathbb{R} \to \mathbb{R}$ is differentiable on $\mathbb{R}$. Then $g(x)$ has one and only one real root if $g'(x)\leq k <0$.
Proof attempt:
Let us assume the contrary, i.e. $g(x)$ has no real zero at all.
Therefore, being continuous, $g(x)... | We can prove that:
$$\lim_{x \to -\infty} g(x) = \infty \quad (1)$$
$$\lim_{x \to \infty} g(x) = -\infty \quad (2)$$
Lets prove the second statement, the first statement can be proved using a similar argument.
Let's assume the opposite, that:
$$\lim_{x \to \infty} g(x) = C \neq -\infty$$
Now if we consider the follo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3241970",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Discontinuous point for a function $\frac{|\sin{x}|}{\sin{x}}$ I want to determine what type of discontinuity a function has by using one-sided limits for the function $$f(x) = \frac{|\sin{x}|}{\sin{x}}$$
I found the left and right hand limits at $x=0$ (because the $f(x)$ is undefined for $f(0)$). I have found that $$\... | It might prove helpful to visualize the function:
At every integer multiple of $\pi$, there is a discontinuity, since $\sin(k\pi) \equiv 0$ for all integers $k$. We make a jump because the function effectively "flips sign" here: wherever $\sin(x)<0$ we have $f(x) = |\sin(x)|/\sin(x) = -1$ and similarly for $\sin(x) > ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3242055",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Given three postive numbers $a,b,c$ so that $a\geqq b\geqq c$. Prove that $\sum\limits_{cyc}\frac{a+bW}{aW+b}\geqq 3$ .
Given three postive numbers $a, b, c$ so that $a\geqq b\geqq c$. Prove that
$$\sum\limits_{cyc}\frac{a+ b\sqrt{\frac{b}{c}}}{a\sqrt{\frac{b}{c}}+ b}\geqq 3$$
I make it
Firstly, we need to have one... | Let $a=x^2$, $b=y^2$ and $c=z^2$, where $x$, $y$ and $z$ are positives.
Thus, we need to prove that
$$\sum_{cyc}\frac{a+\sqrt{\frac{b}{c}}b}{\sqrt{\frac{b}{c}}a+b}\geq3$$ or
$$\sum_{cyc}\frac{x^2z+y^3}{x^2y+y^2z}\geq3.$$
Now, by AM-GM
$$\sum_{cyc}\frac{x^2z+y^3}{x^2y+y^2z}\geq3\sqrt[3]{\frac{\prod\limits_{cyc}(y^3+x^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3242218",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Peculiar (convergent?) definite integral I have been trying to calculate the integral:
$$\int_1^{\infty} \left(\frac{x^2}{\sqrt{x^4-1}}-1\right)dx$$
A hint is to multiply the whole integral by $x^{\lambda}$, calculate the two terms independently as a function of $\lambda$ and then set $\lambda=0$. But this did not work... | The integral in $u\in[0,1]$ is for me simpler, so let us introduce for a handy notation
$$y=y(u) = \sqrt{1-u^4}\ .
$$
Then for the integral to be calculated we observe first
$$
\frac\partial{\partial u}
\left(\frac{1-y}u\right) =
\frac{u^2}y-\left(\frac 1y-1\right)\frac 1{u^2}\ .
$$
So we need to calculate
$$
\begin{a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3242614",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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Why is A union B also called "A or B"? In A union B, the element either belongs to A or B, or A and B right?
So shouldn't it be called A and/or B? Due to this I am unable to solve a problem in my textbook.
| Because the elements of $A\cup B$ are exactly those objects that either belong to $A$, or belong to $B$. The claim "$p$ or $q$" is true if either $p$ is true, or $q$ is true, or $p$ and $q$ are both true.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3242699",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 7,
"answer_id": 5
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Why is the Killing form of $\mathfrak{g}$ restricted to a subalgebra $\mathfrak{a} \subset \mathfrak{g}$ not the Killing form of $\mathfrak{a}$? I know that the Killing form of $\mathfrak{g}$ restricted to an ideal $I \subset \mathfrak{g}$ is just the Killing form of $I$.
However, what happens in general if we relax th... | The question 'what happens' is best answered by looking at an example, so the first question is: where do we find examples of subalgebras that are not ideals?
Here is a class of examples. consider the real Lie algebra $\mathfrak{g} = \mathfrak{sl}_n$ of traceless $n$-by-$n$ matrices. It has a Cartan decomposition $\m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3242799",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Derivative function continuous iff partial derivatives continuous
Let $f:\mathbb{R} ^{n}\rightarrow \mathbb{R} ^{m}$ be differentiable.
The derivative function $Df:\mathbb{R} ^{n}\rightarrow L\left( \mathbb{R} ^{n},\mathbb{R} ^{m}\right)$ is continuous in respect to the operator norm $\left\| A \right\|_{L\left( \math... | The idea behind all the proofs I've seen is to use the mean value theorem (or mean value inequality if you're working in general Banach spaces). This is carried out in a clear fashion in Henri Cartan's book Differential calculus in proposition 3.7.2. BTW this book is out of print, but I think there is a reprint under ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3242905",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Estimating quality of projection Suppose we are given a vector $v$ and vectors $\mu_i$:
$v = \mu_1+\mu_2+...+\mu_m$, where $\mu_i \in R^n$, all $\mu_i$ are of unit length.
Oracle will give me $k$ vectors $\mu_{j_1}, \mu_{j_2},...\mu_{j_k}$ from the original set such that when I project $v$ onto subspace spanned by thes... | In general, the answer is negative. Indeed, assume that $m\leq n-1$, $\alpha\le \sqrt{2}$ and for each $i\le m$, $\mu_i=\sqrt{1-\tfrac{\alpha^2}{2}}e_{m+1}+\tfrac{\alpha}{\sqrt{2}}e_i$, where for each $j$, $e_j$ is $i$-th standard ort of the space $\mathbb{R}^n$ (that is its $i$-th coordinate is $1$ and other coordinat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3243045",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
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What is the difference betwen equivalence and isomorphism of functors in categories. I am learning category theory using Basic Category Theory by Tom Leinster as my main source. In the chapter on natural transformations he says that isomorphism of categories is unreasonably strict for the notion of the sameness of two ... | Well, the actual difference between the two statements is that for an equivalence of categories, we only require that that the composites $F \circ G$ and $G \circ F$ are naturally isomorphic to the identity functors rather than exactly equal. That is, there's a collection of isomorphisms $\eta_x :GF(x) \rightarrow x$ f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3243243",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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Calculate $\int ^{4\pi} _{-4\pi} \frac{(\sin x)^2-(\sin x)^4}{1-(\sin x)^4}dx$
Calculate $$\int ^{4\pi} _{-4\pi} \frac{(\sin x)^2-(\sin x)^4}{1-(\sin x)^4}dx$$
I tried to do this task in several ways, but none of them proved to be effective. For example:
$$\int ^{4\pi} _{-4\pi} \frac{(\sin x)^2-(\sin x)^4}{1-(\sin x... | Hint:
Bioche's rules say you should set $t=\tan x$. Indeed, with some trigonometry,
$$\frac{\sin^2x}{1+\sin^2x}=\frac{\cfrac{t^2}{1+t^2}}{1+\cfrac{t^2}{1+t^2}}=\cfrac{t^2}{1+2t^2},\qquad\mathrm dx=\frac{\mathrm dt}{1+t^2},$$
so the indefinite integral becomes
$$\int\frac{t^2\,\mathrm dt}{(1+2t^2)(1+t^2)}$$
Can you co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3243347",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 3
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Use of the Leibniz integral rule in Laplace transform proof My Laplace transform textbook presents the following theorem:
If $\mathcal{L}\{ F(t) \} = f(s)$, then $\mathcal{L}\{ t F(t) \} = - \dfrac{d}{ds}f(s)$ and in general $\mathcal{L}\{ t^n F(t) \} = (-1)^n \dfrac{d^n}{ds^n} f(s)$.
The proof then begins as follows... |
The Leibniz Rule for an infinite region
If there is a positive function $g(x, y)$ that is integrable, with respect to $x$, on $[0,∞)$, for each $y$, and such that $|\frac{∂f}{∂y} (x, y)| ≤ g(x, y)$ for all $(x, y)$, then
$$\frac{d}{dy}\int_{0}^\infty f(x,y)\,dx=\int_{0}^\infty \frac{\partial}{\partial y} f(x,y)\,dx$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3243542",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Characteristic polynomial of $2\times2$ matrix $A$ with $A^2v=-v$ This is a multiple select question, i.e., more than one answer can be correct:
If $A\ne0$ is a $2\times2$ real matrix and suppose $A^2v=-v$ for all vectors $v\in\Bbb R^2$, then
*
*$-1$ is an eigenvalue of $A$,
*The characteristic polynomisal of $A$ i... | As the rest is easy, I'll just indicate an alternative way to get at the characteristic polynomial, without using either the Cayley-Hamilton theorem or complex numbers. There can be no real eigenvalues, as $A^2v=-v$ for en eigenvector $v$ shows the corresponding eigenvalue$~\lambda$ should have $\lambda^2=-1$, which it... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3243654",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Show that $a_0\leq \alpha \le a_0 +1$ It is given that $$\alpha=[a_0;a_1,a_2,...,a_n]$$ where $a_0,...,a_n$ are all positive integers.
We need to show that
$$a_0\leq \alpha \le a_0 +1$$
My question is when the equality holds ? I guess the question is wrong ... there can't be the equality sign in the question as these ... | The question isn't wrong. Note that it is true, for example, that $2\leq 2.5\leq 3$. It is also true that $2<2.5<3$. Just because the strict inequality is true does not mean the inequality with the equal signs is false. In fact, the strict inequality implies the one with equal signs. Indeed, the statement is true becau... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3243786",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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$\frac{4^p - 1}{3}$ is a Fermat pseudoprime with respect to 2 I have to prove that $n = \frac{4^p - 1}{3}$ is a Fermat pseudoprime with respect to $2$ when $p \geq 5$ is a prime number. I have proved that $n$ is not prime because $4^p - 1 = (2^p-1)(2^p+1)$ and $(2^p + 1)$ is divisible by $ 3$. But now I can't show that... | $n=\dfrac{4^p-1}3=\dfrac{2^{2p}-1}3,$ so $n\mid 2^{2p}-1,\,$ so $\,\color{#c00}{2^{2p}\equiv 1}\pmod{\!n}$
Further, $2p$ divides $2\times\dfrac{(2^{p-1}-1)}3\times{(2^{p}+2)}=\dfrac{2^{2p}-4}3=n-1,\,$ so $n-1 = 2pk$
Therefore, $\!\bmod n,\,$ we have $\,2^{n-1}\equiv (\color{#c00}{2^{2p}})^{k}\equiv \color{#c00}1^k\equi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3243884",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to prove : $\cos^32\theta + 3\cos2\theta = 4(\cos^6 \theta -\sin^6 \theta)$ How to prove : $\cos^32\theta + 3\cos2\theta = 4(\cos^6 \theta -\sin^6 \theta)$
| $4(\cos^6\theta-\sin^6\theta)$
$=4((\cos^2\theta)^3-(\sin^2\theta)^3)$
$=4(\cos^2\theta-\sin^2\theta)(\cos^4\theta+\sin^4\theta+\cos^2\theta\sin^2\theta)$
$=4\cos 2\theta[\{(\cos^2\theta+\sin^2\theta)^2-2\cos^2\theta\sin^2\theta\}+\cos^2\theta\sin^2\theta]$
$=4\cos 2\theta[\{1-2\cos^2\theta\sin^2\theta\}+\cos^2\theta\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3244039",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is an integrated Wiener process recurrent or transient? Like the title says, if I take an integrated Wiener process / Brownian motion $\int ^t _0 W_s ds$, will it be recurrent or transient? Or, under what conditions will it be one or the other?
I know that, for any $t$, the integral is a normal variable ~$N(0,\frac{t^... | It is recurrent.
Let $X_t := \int_0^t W_s \,ds$. I claim that $\varlimsup_{t\to\infty}X_t=\infty$ and $\varliminf_{t\to\infty}X_t=-\infty$, so every real number is visited infinitely often a.s.
Here is a sketch of a proof.
Write $\mathscr F_t := \sigma(W_s ; s \le t)$. For a finite stopping time $T$, write $X_t = X_{t ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3244239",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Question about equilibrium points of non-linear system of ODEs. So I have the system
\begin{align}
x'&=y\\
y'&=-x-y\ln(x^2+4y^2)
\end{align}
To find the equilibrium points I need $x'=0$ and $y'=0$, thus I obtain
\begin{align}
y&=0\\
-x-y\ln(x^2+4y^2)&=0
\end{align}
I don't see how to proceed here. If $y=0$ in the secon... | Let $V((x,y)) = {1 \over 2} (x^2+y^2)$, and $\phi(t) = V((x(t),y(t)))$. Note that
$\phi'(t) = -y^2 \ln(x^2+4y^2)$.
Let $A= \{ (x,y) | {1 \over 4} \le x^2+y^2 \le 4 \}$.
Note that $\phi'(t) \ge 0$ if $x(t)^2+y(t)^2= {1 \over 4}$.
Note that $\phi'(t) \le 0$ if $x(t)^2+y(t)^2= 4$.
$A$ contains no equilibrium points.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3244366",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
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Prove $f(x) = \sum_{n=1}^{\infty} \frac{x^n}{2^n} \cos{nx}$ is differentiable at $(-2, 2)$. Prove $$f(x) = \sum_{n=1}^{\infty} \frac{x^n}{2^n} \cos{nx}$$ is differentiable at $(-2, 2)$.
I can't use formula for radius of convergence, because it's not a power series ($x$ is present also in $\cos$).
| HINT: Try to prove uniform convergence on $[-2+\alpha,2-\alpha]$ for any $0<\alpha<2$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3244502",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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- Show the set $A=\{(m,n)\in N\times N : m\leq n\}$ is countably infinite.
*
*Show the set $A=\{(m,n)\in N\times N : m\leq n\}$ is countably infinite.
If $A$ is countable then we need to show that there is a bijection between $A$ and $\mathbb{N}$, but how can I show $A$ is countably infinite?
Thanks...
| Note that $A$ can be written as
$$
A = \bigsqcup_{n \in \mathbb{N}}\{(m,n) : m \leq n\}.
$$
That is, $A$ is the disjoint countable union of sets $F_n = \{(m,n) : m \leq n\}$. These are finite: for a fixed $n$, the set $F_n$ has $n$ elements, namely $(1,n) , (2,n) \dots, (n,n)$. Thus $A$ is a countable union of countab... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3244649",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Quadratics vs Hermitian forms The quadratic form is given by:
$$Q(\mathbf{x}) =\langle\ \mathbf{x} \ | \ A\mathbf{x} \rangle = x^TAx$$
Where $Q$ is a real scalar and hence $Q = Q^T$
The hermitian form is given by:
$$H(\mathbf{x}) = \langle \ \mathbf{x} \ | \ A\mathbf{x} \ \rangle=x^{\dagger}Ax$$
Where $H$ is also scala... | For $Q$ to be a quadratic form $A$ has to be a symmetric matrix, and for $H$ to be a Hermitian form, $A$ has to be Hermitian i.e. $A = \overline{A^T}$. They are the same over real numbers, but if you work over the complex numbers of course they would be different.
An important reason why we usually use Hermitian forms... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show the series $\sum\limits_{n=1}^\infty\frac{1}{2+\sqrt{n}}$ diverges I know intuitively why this series diverges but I can't really get a proof.
So I am trying to use that the fact that: $\sum\limits_{n=1}^\infty\frac{1}{2+\sqrt{n}} > \sum\limits_{n=1}^\infty\frac{1}{2+{n}} $.
And then from there I want to get it d... | You already have the answer. $\sum_{n=1}^\infty\frac1{2+n}$ is a tail of the divergent harmonic series and thus diverges itself. The original series therefore diverges by the limit comparison test.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3244930",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to solve $y'^2 +yy'+x=0$? I encountered this ODE while looking for a curve which is orthogonal to the family of lines given by $$ y = mx + \frac{1}{m} \ \ \ m \in \Re \ \ ...[1] \\ $$
I setup an ODE for [1] by puting $m = y'$ in [1],
$$ \ \ y = xy' + \frac{1}{y'} \ \ ...[2]. \\ $$ Next, to get a family of ortho... | Writing your equation in the form
$$y(x)=-y'(x)-\frac{x}{y'(x)}$$ differentiating this equation with respect to $x$
$$\frac{d}{dx}(y'(x))=\frac{y'(x)^3+y'(x)}{x-y'(x)}$$
substituting
$$v(x)=y'(x)$$
$$x'(v)=-\frac{v^2}{v^3+v}+\frac{x}{v^3+v}$$
Calculating
$$\mu(x)=\int e^{1/(v^3+v)}dx=\frac{\sqrt{v^2+1}}{v}$$ so we get
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3245060",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Interesting primary school problem Let say
$$\frac{a}{b+c}+\frac{d}{e+f}+\frac{g}{h+i}=1$$
Given that $$a,b,c,d,e,f,g,h,i$$ represents number 1,2,3,4,5,6,7,8,9 (we don't know which alphabet represent which digit)
When dealing with this problem, I came up with the following question:
Q1)Is that an algebraic way to solv... | The one algebraic approach I can think of is to render
$(1/2)+(1/3)+(1/6)=1$
and thereby identify
$a/(b+c)=(1/2)$
$d/(e+f)=(1/3)$
$g/(h+i)=(1/6)$
Now there are only a few possibilitues for the $1/6$ fraction because the denominator has to be less than $9+9=18$ and has to be a multiple of $6$. Only the following fit bo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3245149",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Real Analysis, Integration
Let $f:\mathbb{R}^{n}\rightarrow \mathbb{R}$ be a continuous function such that $\lim_{|x|\rightarrow \infty} f(x)=0$. Prove that $\lim_{k\rightarrow \infty}\int_{[0,1]^{n}}f(kx)dx=0. $
I don't know how to proceed, anyone have any idea? Thanks!!
My idea: If $\lim_{|x|\rightarrow \infty} f(x... | Hint: Since $f$ is continuous and $\lim_{\lvert x\rvert\to\infty}f(x)=0$, we have $f$ is bounded and uniformly continuous. Now what is the obvious thing we can do to $\int_{[0,1]^n} f(kx)\,\mathrm{d}x$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3245299",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Proving that the $\frac {\xi +\zeta\eta}{\sqrt {1+\zeta^2}}$ has normal distribution (0,1) The task: $\xi, \eta, \zeta \sim N(0,1)$ and independent. Prove, that $\frac {\xi +\zeta\eta}{\sqrt {1+\zeta^2}} \sim N(0,1).$ (1)
It is clear, that with fixed $\zeta$ we get, that (1) has expected value = 0 (as the sum of normal... | Let $R=\frac{\xi+\zeta\eta}{\sqrt{1+\zeta^2}}$. You want to show $E\exp(itR)=\exp(-t^2/2)$. Write $E\exp(itR)=E(E[\exp(itR)|\zeta])$. The inner, or conditional expectation is $$\begin{align*}E[\exp(itR)|\zeta]&=\tag{*}
E\exp(it\xi/\sqrt{1+\zeta})|\zeta) \times E\exp(it\eta/\sqrt{1+\zeta})|\zeta)\\
&= \exp(-\frac{t^2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3245400",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Are $m\mathbb{Z} \cong n\mathbb{Z}$ as rings for arbitrary $m,n \in \mathbb{N}$? Are $m\mathbb{Z} \cong n\mathbb{Z}$ as rings for arbitrary $m,n \in \mathbb{N}$?
$\alpha: m\mathbb{Z} \rightarrow n\mathbb{Z}$
$\alpha(m)=n$.
Then $\alpha(ma) = \alpha(mb) \rightarrow a=b$ so $\alpha$ is injective.
It can also be shown th... | It is not ring isomorphic. I will give a particular case and you will easily develop for general case. Take $n=2$ and $m=3$ and consider $f:3 \Bbb Z \to 2\Bbb Z$. If it were a ring isomorphism, then it maps generator to generator. So assume for example, $f(3)=2$ Now $$f(9)=f(3+3+3)=2+2+2=6$$ and $$f(9)=f(3 \times 3)=f... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to derive the Method of Moments estimator of mu, using the second moment of X, when X is norm distributed? If $X_1,\ldots,X_n$ follow a normal distribution, where the variance $\sigma$ is given, how can you derive the MME of the mean $\mu$ using the second moment?
| Note that the second moment of $X$ is
$$E[X^2] = \sigma^2 + \mu^2,$$
so by subtracting the known variance from the second moment, you can find the square of the mean.
To this end, compute an estimate of the second moment
$$\hat s = \frac {1}{N} \sum_{i=1}^N X_i^2,$$
then find the estimate of the mean with
$$\hat \mu = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3245624",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the roots of equation based on some geometry hints Plots of the equations $y = 8 - x^2$ and $|y|=\sqrt{8+x}$ are symmetric w.r.t. the line $y=-x$. We have to solve the equation $$8-x^2=\sqrt{8+x}$$
| If $y=8-x^2$ and $|y|=\sqrt{8+x}$, then $y^2-y=(8+x)-(8-x^2)=x+x^2,$
so $y(y-1)+x(-1-x)=0,$ so $y(y-1-x)+x(y-1-x),$ so $(y+x)(y-1-x)=0,$
i.e., $y=-x$ or $y=1+x$. Therefore $x$ must be a solution of $8-x^2=-x$ or $8-x^2=1+x$.
Can you solve these quadratic equations? Note that you want solutions where $y=8-x^2\ge0$. ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Check my quick proof $P_1 \oplus P_2$ is projective $\iff P_1,P_2$ are projective. Just want to check if I am right in the $\Longrightarrow$ direction. This is Exercise 3 from Dummit Foote.
($\Longleftarrow$) Given SES $0 \to L \to M \to N \to 0$, $ Hom(P_1, -) \oplus Hom(P_2,-) = Hom(P_1 \oplus P_2, -)$. Take the dir... | To show $P_1$ is projective you need to show that any map $f:P_1 \to N$ factors through M, that is, show there exists a map $g: P_1 \to M$ such that $\phi g = f$. So your idea to involve the direct sum is right but you need to switch up the order of operations a bit.
An alternative would be to use the fact that a modul... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Ball around a set equals union of balls around each of its points I have this problem:
Let $(X,d)$ be a metric space and let $A \subseteq X$. Define for any $ \epsilon > 0$:
$$B_d(A,\epsilon) = \left\{ x \in X \ : \ d(x,A) < \epsilon \right\}.$$
Show that $B_d(A,\epsilon) = \bigcup_{x \in A} B_d(x,\epsilon)$.
I... | Filling the minor gap in the forward inclusion:
Take $p \in B_d(A,\varepsilon)$. So $d(p,A) < \varepsilon$. Suppose that for all $x \in A$ we would have that $\varepsilon \le d(x,p)$.
This means that $\varepsilon$ is an lowerbound for the set $\{d(x,p): x \in A\}$ and $d(p,A)= \inf \{d(x,p): x \in A\}$ is the maximum o... | {
"language": "en",
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"source": "stackexchange",
"question_score": "2",
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dimension of a connected Manifold Currently, I am studying smooth manifolds and I want to solve some exercises. There is a question that says:
Show that the dimension of a connected topological manifold is defined without ambiguity! Meaning that if $\dim M=n$, with the change of charts, the dimension still is $n$. Then... | If $M$ is a smooth manifold of dimensions $m$ and $n$ then for each $p\in M$, we can find charts about $p$ with images open in $\mathbb{R}^n$ and $\mathbb{R}^m$. Hence $T_pM$ is isomorphic to $\mathbb{R}^n$ and $\mathbb{R}^m$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Suppose that $G$ is a group of order $924=2^2\cdot3\cdot7\cdot 11$. Prove that $G$ has an element of order $77$.
Suppose that $G$ is a group of order $924=2^2\cdot3\cdot7\cdot 11$. Prove that $G$ has an element of order $77$.
My attempt:
By Sylow theorems, we know that there exist elements $ a, b\in G$ with $o(a)=7 ... | Well we know there are elements of order $7$ and $11$ and if any pair of such elements commute then they generate a cyclic subgroup of order $77$.
I think you can argue that if the number of subgroups of order $11$ is not $1$ then it is $12$ (Sylow again: $\equiv 1 \bmod 11$). Take these two cases together.
Take an ele... | {
"language": "en",
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"source": "stackexchange",
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$\int_0^{100}\frac{e^{-x}}{x+100}dx>0.005$? $\int_0^{100}\frac{e^{-x}}{x+100}dx>0.005$?
My attempt: $$\int_0^{100}\frac{e^{-x}}{x+100}dx>\int_0^{100}\frac{e^{-x}}{200}dx=\frac{1-e^{-100}}{200}$$ A little bit error. How to amend it?
| The idea: The integrand decreases rapidly on the given interval, so the idea is to estimate the integral from below by integrating over a shorter interval $[0, a]$, and then continue with your approach, but with a better bound for the denominator:
For $0 < a < 100$ we can estimate
$$I = \int_0^{100}\frac{e^{-x}}{x+100... | {
"language": "en",
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Equivalence of completeness statements I need to "find a metric d such that (0,1) is complete". Is it equivalent to instead find a homeomorphism f from (0,1) to a complete space, so for instance $f=\tan(-\frac{\pi }{2}+\frac{\pi x}{2})$, and then say $d(x)=|f(x)-f(y)|$
| Let $X$ be topological space and $M$ metric space, $f\colon X\to M$ a homeomorphism.
Define $d_X(x,y):=d_M(f(x),f(y))$. You can easily check that this is a metric on $X$.
Also, $f$ becomes isometry, so $(x_n)$ is a Cauchy sequence in $X$ iff $(f(x_n))$ is Cauchy sequence in $M$ and by continuity $(x_n)$ converges in $... | {
"language": "en",
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"source": "stackexchange",
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$\cos (n \phi)$ in terms of $\cos (\phi)$ Consider the following quantity:
$$\cos(n \phi), \ n \in \mathbb{N}, \ n > 1$$
I know it is possible to alternatively express it as a polynomial of degree $n$, with powers of $\cos (\phi)$ and $\sin (\phi)$.
But is it possible to express $\cos(n \phi)$ in terms of $\cos(\phi)$ ... | No. The right-hand side asds up to one big $Z\cos(\phi+\omega)$.
$$A\cos(\phi+\alpha)+B\cos(\phi+\beta)+\ldots\\
=\cos\phi(A\cos\alpha+B\cos\beta+\ldots)-\sin\phi(A\sin\alpha+B\sin\beta+\ldots)\\
=P\cos\phi+Q\sin\phi=Z\cos(\phi+\omega)$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Number of binary words that can be formed
How many binary words of length $n$ are there with exactly $m$ 01 blocks?
I tried by finding number of ways to fill $n-2m$ gaps with $0$ and $1$ such that no $'01'$ block gets created again. But this method is not working and I am stuck in this problem. Please provide me an e... | Here is another approach. It is "relatively easier" in the sense that setting up recurrence relations that condition on the last few digits of the binary word to force what we're interested in, is a standard way to approach such questions.
Let $a_{n,m}$ be the number of binary words of length $n$ with exactly $... | {
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Connected, locally compact, paracompact Hausdorff space is exhaustible by compacts I'm trying to understand the proof of the following:
A connected, locally compact, paracompact Hausdorff space $X$ has an exhaustion by compact sets,
That is, there exists a sequence $(K_n)_n$ of compact subsets of $X$ whose union is $... | One starts by noting that all the set of all open $O$ with $\overline{O}$ compact is an open cover of $X$ by locally compactness and Hausdorffness.
The paracompactness of $X$ then gives us a locally finite refinement $(U_i)_{i \in I}$ of that cover. It's not the $U_i$ that need to be compact in this proof, but their c... | {
"language": "en",
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Find all numbers x which satisfy $|x^2 + 2| = |x^2 − 11|$. This question is taken from book: Exercises and Problems in Calculus, by
John M. Erdman, available online, from chapter 1.1, question $4$.
Request help, as not clear if my approach is correct.
(4) Find all numbers x which satisfy $|x^2 + 2| = |x^2 − 11|$.
Ha... | Hint:
For real $x,$ $x^2\ge0$, so $x^2+2>0$, so $|x^2+2|=x^2+2$. $|x^2-11|=x^2-11$ or $-(x^2-11).$ Equate them and see what you get.
| {
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"source": "stackexchange",
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What is the probability that P and Q have no common elements? A is a set containing n elements. A subset P of A is chosen at random. The set A is reconstructed by replacing the elements of the subset of P. A subset Q of A is again chosen at random. Find the probability that P and Q have no common elements.
I tried to c... | $P,Q$ can both be one of the $2^n$ subsets of $A$. The total number of ways to form $P$ and $Q$ is $2^n\cdot2^n=2^{2n}$.
| {
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Evaluate $\int_0^{\infty} \frac {\ln(1+x^3)}{1+x^2}dx$
Prove that $$\int_0^{\infty} \frac {\ln(1+x^3)}{1+x^2}dx=\frac {\pi \ln 2}{4}-\frac {G}{3}+\frac {2\pi}{3}\ln(2+\sqrt 3)$$ Where $G$ is the Catalan's constant.
Actually I proved this using the Feynman's trick namely by introducing the parameter $a$ such that $$\x... | Note
$$\int_0^{\infty} \frac {\ln(1+x^3)}{1+x^2}dx
= \underset{= \frac\pi4\ln2+G}{ \int_0^{\infty} \frac {\ln(1+x)}{1+x^2}dx}
+ \underset{=K}{\int_0^{\infty} \frac {\ln(1-x+x^2)}{1+x^2}dx}
\tag1
$$
To compute $K$, let
$J(a)= \int_0^{\infty} \frac {\ln\left(\frac12 (1+x^2)\sec a-x\right)}{1+x^2}dx
$
$$J’(a) = \int_0^\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How many series of length $n$ combined from the numbers $0,1,3,4$ , that don't contain the sequences $04,40,13,31$ or $0X0, 1X1, 3X3, 4X4$ are there? How many series of length $n$ combined from the numbers ${0,1,3,4}$ , that don't contain the sequences $04,40,13,31$ or $0X0, 1X1, 3X3, 4X4$ (where X can be any number fr... | As soon as I have a series of length $2$ or more it will end with a pair $\dots ab$
To add $c$ so we get $\dots abc$ we need to make sure that $c\neq a, 4-b$. This excludes two possibilities unless $a=4-b$, but in this case the sequence was already bad, so we can exclude it.
(This can also be used to analyse what happe... | {
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How many labeled trees of $n$ vertices exist which have at most $k$ degree of any vertex? I am interested how one can count all the trees with $n$ vertices from all $n^{n-2}$ trees, that have at most $k$ degree of a vertex. Is there a way to do it for any $k$?
All trees with degree at most $k$ can be encoded to Prufer ... | There is no "closed form" solution in terms of elementary functions. However, the number of lists of length $n-2$ with entries in $\{1,2,\dots,n\}$ where each element appears at most $k-1$ times can be written as
$$
(n-2)!\cdot[x^{n-2}]\Big(1+x^1/1!+x^2/2!+\dots+x^{k-1}/(k-1)!\Big)^n
$$
Here, $[x^m]f(x)$ refers to the ... | {
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Find $9$'th derivative of $\frac{x^3 e^{2x^2}}{(1-x^2)^2}$ How can I find $9$'th derivative at $0$ of $\displaystyle \frac{x^3 e^{2x^2}}{(1-x^2)^2}$. Is there any tricky way to do that?
This exercise comes from discrete mathematic's exam, so I think that tools like taylor can't be used there.
| Hint: Take $\ln$ of both sides of
$$y=\frac{x^3 e^{2x^2}}{(1-x^2)^2}$$
and use these facts that
$$\dfrac{d^n}{dx^n}\ln(1+x)=\dfrac{(-1)^{n-1}(n-1)!}{(1+x)^{n+1}}$$
$$\dfrac{d^n}{dx^n}\ln(1-x)=\dfrac{(n-1)!}{(1-x)^{n+1}}$$
| {
"language": "en",
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"source": "stackexchange",
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Slot machines fundamental matrix interpretation The problem:
A man is playing two slot-machines. The first machine pays off with probability c the second with probability d. If he loses, he plays the same machine again. If he wins, he switches to the other machine. Let S, be the state of playing the i-th machine. Creat... | The fundamental matrix of an ergodic Markov chain is
$$Z = \left(I-P+W\right)^{-1}$$
where $P$ is the transition matrix and $W$ is a matrix where each row is the fixed probability vector $w = (w_i)$ (the limiting distribution).
You can calculate the mean first passage times $m_{ij}$ (expected number of steps to reach s... | {
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Is this solution correct? I thought it was meant to be $y = 1/t-c_i,$ if the solution is wrong, what's the correct answer? The question is solve the following ODE: $t^2y'' =(y')^2 ,\; t>0.$
Here's the solution file:///M:/pc/My%20Documents/Doc3.pdf
This is the solution my lecture provided and I think there's an error s... | Hint: Substitute $$y''=v'$$ then you will get $$\frac{t^2v'(x)}{v(x)}=1$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How many different ordered pairs of 20 elements? Question: How many different ordered pairs of 20 elements $\left( x_{1}, x_{2}, x_{3}, \dotsc, x_{20} \right)$ can you create if $x_{1}, x_{2}, x_{3}, \dotsc, x_{20}$ are non-negative integers and $x_{1} \leq 3, \, x_{2} \leq 6, x_{3} \leq 9, \dotsc, x_{20} \leq 60$ and ... | The number of choices that you have for the $k^{th}$ number is $3k+1-(k-1) = 2(k+1)$, because $3k+1$ is the total number of choices and $k-1$ are the numbers that have already been selected and are excluded from the available choices.
Multiplying these for $k$ ranging from $1$ to $20$ we get the desired result. Namely,... | {
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Prove that $\langle f(x) \rangle $ is prime ideal in $\Bbb Z[x]$ if $f(x)$ is irreducible over $\Bbb Z$ I know $(x^2+1)$ is prime ideal of $\Bbb Z[x]$ without being maximal ideal. It can be easily proved as quotient ring isomorphic to integral domain $\Bbb Z[i]$. My question is the generalization of this i. e. if $(... | A simple argument could be:
*
*If $f(x)$ is irreducible in $\Bbb{Z}[x]$, including that it has no non-unit constant factor, then it is also irreducible in $\Bbb{Q}[x]$ (the usual argument invokes Gauss's lemma at a key step).
*Therefore $f(x)$ generates a maximal ideal in $\Bbb{Q}[x]$.
*Therefore $\Bbb{Q}[x]/\lang... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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lengths of sides in golden ratio isosceles triangles The figure below shows three different isosceles triangles. every triangle is either 36-36-108 or 36-72-72. The base of the outermost triangle has length $\phi$.
Find the lengths of both lines AT and MT.
Can someone please help me figure this out? What I'm thinking... | As shown on Wikipedia's article about golden ratio (letters in the blockquote indicate points on the picture provided by Wikipedia):
If angle BCX = α, then XCA = α because of the bisection, and CAB = α because of the similar triangles; ABC = 2α from the original isosceles symmetry, and BXC = 2α by similarity. The an... | {
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Unit vector notation - both line and hat? I know that unit vectors are usually written in bold with a hat for example: $$\hat{\mathbf{i}}$$
But if you use vector notation with arrows (as I do when I write with pen and papper), should you both have an arrow above the vector and a hat, or just the hat?
Thanks!
| In general, what indicates whether a vector is unit vector or not is the letter used rather than the hat ($\hat{}$). So, as long as you prevent confusion by not giving name $i,j,k,e_1,e_2$ etc. to some other vectors, $\vec{i}$ still is a unit vector. But I should also note that this also depends on the context in which... | {
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Proving monotone function of two variables is integrable
Let $f:[0,1]^2\rightarrow \mathbb R$ be a monotone function of two variables, that is, $x\leq x'$ and $y\leq y' \implies f(x,y)\leq f(x',y').$ Prove that $f$ is Riemann integrable.
I want to "copy" and generalize the argument for the one dimensional case. Well,... | The key idea is that you have $N^2$ squares each of area $\dfrac{1}{N^2}$, but after telescoping the sum "as much as possible", there are only $2N-1$ summands left, each summand being of the form $f(p) - f(q)$; which is bounded by $f(1,1) - f(0,0)$. Let $S$ denote an arbitrary subrectangle determined by the partition $... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why does this inequality hold for smallest positive non-square modulo p? Let $p$ be an odd prime and $q$ the smallest positive integer that is not a square modulo $p$. Show that $q<\sqrt{p}+1$?
I can show that $q$ also has to be prime and that $p$ is a divisor of $q^\frac{p-1}{2}+1$. But I am not sure how this can help... | Assume $q>\sqrt p+1$. Let $a=\left\lceil\frac pq\right\rceil$. Then
$$1\le a\le\left\lceil\frac p{\sqrt p+1}\right\rceil
=\left\lceil\sqrt p-1+\frac 1{\sqrt p+1}\right\rceil\le\left\lceil\sqrt p\right\rceil<q,$$
hence $a$ is a square $\bmod p$ and so $aq$ is a non-square.
But
from $p\le aq<p+q$, we see that $aq\bmod ... | {
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Do the categories of Sets and Smooth manifolds with smooth functions have zero morphisms. And how are compositions with the empty set defined? When reading wiki zero morphisms it seems that the category of sets does not have zero morphisms. Also, I could not find how the composition with the empty map works in the Set.... | Thomas Andrews has answered your main question about zero morphisms, so I'll just answer the question "how are compositions with the empty set defined?"
In the category of sets, a function $X\to Y$ is a set $f\subseteq X\times Y$ such that for all $x\in X$, there is a unique $y\in Y$ such that $(x,y)\in f$. From this d... | {
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Uniqueness of $\mathbb{R}$ In the very beginning of baby Rudin we are given
$\textbf{1.19 Theorem}$ There exists an ordered field $\textit{R}$ with the least upper bound axiom. Moreover, $\textit{R}$ contains $\textit{Q}$ as a subfield.
Are the real numbers the $\textit{unique}$ ordered field with l.u.b. property wi... | Yes. The relevant property is not, however, that $\mathbb{Q}$ is a subfield, but rather the fact of the supremum property (or any one of any number of other statements equivalent to Dedekind completeness). $\mathbb{R}$ is the only ordered field (up to isomorphism) with this property.
| {
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"answer_id": 0
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Problem with change of variables in an integral In calculating the electric potential of a charged line segment of length $2L$ (don't worry, this is pretty much where the physics ends), I encountered the following integral: \begin{equation}V(r,z)=\frac{Q}{8\pi\varepsilon L}\int_{-L}^L\frac{dz'}{\sqrt{r^2+(z-z')^2}}.\en... | Actually, $V(r,z)=V(r,-z)$, as can be easily seen by the substitution $z'\to -z'$ in the integrand, which has the effect of replacing $z$ with $-z$.
Indeed, letting $y=z'$ for ease of notation we have
$$
V(r,z)=c\int_{-L}^L\frac{dy}{\sqrt{r^2+(z-y)^2}}=c\int_{L}^{-L}\frac{-dy}{\sqrt{r^2+(z+y)^2}}=c\int_{-L}^L\frac{dy}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3249065",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Minimum number of times monkey must travel While working on an unrelated subject, I found a problem which could be alternatively be stated as the following:
A monkey must travel a path of length $n-1$, from $a_1$ to $a_n$. On every turn, he may jump any distance forward, but cannot go backwards. He may jump any number... | There was an answer here a few hours ago claiming $n^2/4.$ I did not read it in full, and I do not know why it's been deleted. But I think the claim is correct. The minimum number of travels is ${n^2\over 4}$ for even $n$, and ${n^2-1 \over 4}$ for odd $n$.
(Credits: the necessity argument I stole from the now-delet... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3249281",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
Getting a one to one morphism $p : G \to GL_n(\mathbb{C})$ I am studying representation theory, and I would like to find an algorithm that finds, given a finite group $G$, a one to one morphism $p : G \to GL_n(\mathbb{C})$ (the integer $n$ is also found by the algorithm). I don't necessarily want this algorithm to be e... | Every finite group embeds into a permutation group; this is Cayley's theorem. The proof of the theorem is essentially constructive, with $G$ embedding into $S_{|G|}$. That is, there is an algorithm with input a Cayley table* of a finite group $G$ and output a permutation group $\operatorname{Perm}(G)$ isomorphic to $G$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3249417",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Uniform convergence of $f_n(x) = \left(1 + \frac{x}{n}\right)^n$ when calculating limit Calculate$$
\lim_{n \rightarrow \infty} \int_0^1 \left(1 + \frac{x}{n}\right)^ndx
$$
My attempt - if
$$
f_n(x) = \left(1 + \frac{x}{n}\right)^n
$$
converged uniformly for all $x \in [0,1]$ then I could swap integral with limes and s... | You can use Dini's theorem.
On a compact set $K$, if a sequence of continuous functions $\langle f_n(x) \rangle$
$a)$ is monotone in $n$ for each $x \in K$
$b)$ converges pointwise to a continuous function of $x \in K$
then the convergence is uniform.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3249567",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 1
} |
Find $\lim_{x\to\infty}1+2x^2+2x\sqrt{1+x^2}$ Consider the function
$$f(x)=1+2x^2+2x\sqrt{1+x^2}$$
I want to find the limit $f(x\rightarrow-\infty)$
We can start by saying that $\sqrt{1+x^2}$ tends to $|x|$ when $x\rightarrow-\infty$, and so we have that
$$\lim_{x\rightarrow-\infty}{(1+2x^2+2x|x|)}=\lim_{x\rightarrow-\... | For negative $x$, we have
$$\sqrt{1+x^2}=-x\sqrt{1+\frac{1}{x^2}}=-x\left(1+\frac{1}{2x^2}+O(x^{-4})\right)$$
So we have
$$
\lim_{x\rightarrow-\infty}(1+2x^2+2x\sqrt{1+x^2})=\lim_{x\rightarrow-\infty}\left(1+2x^2-2x^2\left(1+\frac{1}{2x^2}+O(x^{-4})\right)\right)=\\
=\lim_{x\rightarrow-\infty}O(x^{-2})=0
$$
You were m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3249693",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 1
} |
Alternating sum of binomial coefficients multiplied by index to an n-2 extent Doing research in probability modelling I obtained an identity, which is correct for $n > 2$.
$$\sum\limits^n_{i=1}(-1)^{n+i}{{n}\choose{i}}i^{n-2}=0$$
How can it be proven directly? In what literature can I find it?
| This is a variation using the coefficient of operator $[z^n]$ to denote the coefficient of $z^n$ in a series. Recalling $e^z=\sum_{j=0}^\infty \frac{z^j}{j!}$ we can write for instance
\begin{align*}
n![z^n]e^{kz}=k^n\tag{1}
\end{align*}
We obtain for $n>2$
\begin{align*}
\color{blue}{\sum_{k=1}^n}&\color{blue}{(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3249780",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
Show that this function is bounded Let $f$ be a $\mathbb R \rightarrow \mathbb R$ continuous function such that : $\lim_ {x \to \pm \infty} f(x) \in \mathbb R$ and $\lim_ {x \to 0} f(x) \in \mathbb R$
How can one show that $f$ is bounded ? I get it "intuitively" but I cant show it rigorously
| If $\lim_{x\to-\infty} f(x)=a$ and $\lim_{x\to\infty} f(x)=b$ put $|a|+|b|+1=:c$. There is an $M>0$ such that $|f(x)|\leq c
$ for all $x\geq M$ and all $x\leq-M$. Since $f$ is continuous there is a $c'$ such that $|f(x)|\leq c'$ for all $x\in[-M,M]$. It follows that $|f(x)|\leq c+c'$for all $x\in{\mathbb R}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3249892",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
} |
Calculate $\lim_{\epsilon\rightarrow 0^+} \int_0^1 \frac{1}{\epsilon x^3+1} dx$
Calculate $\lim_{\epsilon\rightarrow 0^+} \int_0^1 \frac{1}{\epsilon x^3+1} dx$
I tried to use:
$$\int_0^1 f(x) \le \int_0^1 \frac{1}{1+\epsilon x^3} dx \le \int_0^1 \frac{1}{1+0} dx=1$$However I have a problem to find $f(x)$ such that $\... | Since $\epsilon>0,$ you have $f_\epsilon (x)\leq f(x),$ where $f_\epsilon(x)=\frac{1}{1+\epsilon x^3}$ and $f(x)=1.$ By dominated convergence theorem ($f$ is integrable on $[0,1]$), $f_\epsilon(x)\to f(x)$ as $\epsilon \to 0^+$ so $\int_0^1 f_\epsilon(x)\,\mathrm{d}x\to \int_0^1 f(x)\,\mathrm{d}x=1.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3250083",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Vector form for Taylor series What is the vector form of Taylor series for a vector valued function of a scalar variable $f:\mathbb R\to\mathbb R^n $? I presume it is exactly the same as the classical Taylor series but for confirmation I have been searching the internet to no avail. Can anyone also point out a referen... | If it's a function from $f:\mathbb{R} \rightarrow \mathbb{R}^n$, the Taylor expansion would be exactly similar expansion for $f:\mathbb{R} \rightarrow \mathbb{R}$.
Let the $f:\mathbb{R} \rightarrow \mathbb{R}^n$ be equal to $f\left(x\right)=(g_1\left(x\right),...,g_n\left(x\right))$
Consider the Taylor expansion for ea... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3250302",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Derivative of a (complicated) sum Given
$$
f(x)=e^{-ax}\ \sum_{k=0}^{r-1}\frac{(ax)^k}{k!}
$$
How do I show that
$$
f'(x)= -\frac{a^r}{(r-1)!}x^{r-1}e^{-ax}
$$
Thank you in advance!
| It is:
$$f(x)=e^{-ax}\ \sum_{k=0}^{r-1}\frac{(ax)^k}{k!}=e^{-ax}\left(e^{ax}-\sum_{k=r}^{\infty} \frac{(ax)^k}{k!}\right)=1-e^{-ax}\sum_{k=r}^{\infty}\frac{(ax)^k}{k!} \Rightarrow \\
f'(x)=\color{red}{ae^{-ax}}\sum_{k=r}^{\infty}\frac{(ax)^k}{k!}-\color{red}{ae^{-ax}}\sum_{k=r}^{\infty}\frac{(ax)^{k-1}}{(k-1)!}=\color{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3250456",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Taylor polynomial of $f(x_1,...,x_m)=\varphi(e^{a\sum_{i=1}^mx_i})$
Let $\varphi:\mathbb{R}\to\mathbb{R}$ be a $C^3(\mathbb{R})$ function, with $a\in\mathbb{R}$. Find the Taylor polynomial of degree $3$, centered in the origin $p=(0,...,0)$, of $f(x_1,...,x_m)=\varphi(e^{a\sum_{i=1}^mx_i})$.
If we define $g=e^{a\sum_... | Replacing $x$ by the polynomial $P_{3,p,g}$ (taylor series of $g(x)$) in $P_{3,g(p),\varphi}$, and ignoring the terms with degree greater than $3$, we obtain the following expression:
$\varphi(1)+\varphi'(1)(a\sum_{i=1}^mx_i+\frac{a^2}{2}\sum_{i,j=1}^mx_ix_j+\frac{a^3}{6}\sum_{i,j,k=1}^mx_ix_jx_k)+\frac{1}{2}\varphi''(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3250584",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
If $f$ is continous at $c$, prove $\lim_{h \to 0} (\inf \,\{f(x)\mid c \leqslant x \leqslant c+h\})=f(c)$(duplicated) Let $f$ be continous at $c$. Prove $$\lim_{h \to 0} \left(\inf \,\{f(x)\mid c \leqslant x \leqslant c+h\}\right)=f(c)$$
This fact is used in Spivak's book to prove 1nd Fundamental Calculus Theorem.
Thi... | Actually, that's not the only thing Spivak uses in his proof. He first considers the case $h > 0$, then $h < 0$, and then claims that $\lim \limits_{h \to 0} m_h = 0$. So, to take into account both signs of $h$, lets do the following: suppose $a < c < b$, and for any $h \in \Bbb{R}$ , define the set
\begin{equation}
A_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3250678",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How to calculate limit as $n$ tends to infinity of $\frac{(n+1)^{n^2+n+1}}{n! (n+2)^{n^2+1}}$? This question stems from and old revision of this question, in which an upper bound for $n!$ was asked for.
The original bound was incorrect. In fact, I want to show that the given expression divided by $n!$ goes to $0$ as $... | $$\frac{(n+1)^{n^2+n+1}}{n! (n+2)^{n^2+1}}$$
= $$\frac{(1+\frac{1}{n})^{n^2+n+1}}{n! (1+\frac{2}{n})^{n^2+1}} \frac{n^{n^2+n+1}}{n^{n^2+1}}$$
=$$\frac{(1+\frac{1}{n})^{n^2+n+1}}{n! (1+\frac{2}{n})^{n^2+1}} \frac{n^nn^{n^2+1}}{n^{n^2+1}}$$
=$$\frac{(1+\frac{1}{n})^{n^2+n+1}}{n! (1+\frac{2}{n})^{n^2+1}} n^n$$
=$$\frac{((... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3250902",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
"Prove that a topology Ƭ on X is the discrete topology if and only if {x} ∈ Ƭ for all x ∈ X" This question is from "Introduction to Topology: Pure and Applied," by Colin Adams and Robert Franzosa.
Here's how the authors define a topology:
Let X be a set. A topology Ƭ on X is a collection of subsets of X, each called a... | You correctly argued one direction -- if it has the discrete topology, the topology is the power set of $X$, so it contains all subsets of $X$, including the singletons $\{x\}$.
For the other direction, use axiom (iii) of a topology, and the fact that every subset of $X$ can be written as a union of singleton sets $\{x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3251008",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Almost quadratic computational complexity Suppose I can bound the running time of my algorithm as $O(a_N N^2)$ for any positive increasing sequence $\{a_N\}$ that diverges to infinity. Does this imply that my algorithm's running time is actually $O(N^2)$?
N.B. I understand that the running time can be bounded by $O(N^{... | Yes it does. Let $t_N$ denote the algorithm time. Let $f_N = \max\{t_M/M^2:M\le N\}$. Suppose $f_N$ diverges. Then $t_N$ fails to be $O(\sqrt{f_N} N^2)$. Therefore $f_N$ does not diverge. Since $f_N$ is a non-decreasing sequence, it must be bounded. Set $C= \sup f_N$. Then $t_N \le C N^2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3251171",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
$\lim_{(x,y)\to 0} \frac{x\sin(y)- y\sin(x)}{x^4 + y^4}$ without polar coordinates? I have the following limit:
$$\lim_{(x,y)\to 0} \frac{x\sin(y)- y\sin(x)}{x^4 + y^4}$$
And I must evaluate it without polar coordinates. I have tried a lot of stuff but nothing works. Can someone give me a hint?
| You can easily disprove that the limit exists by considering the one dimensional family of rays $y=\lambda x, \lambda\in\mathbb{R}$ and taking the limit to the origin along them instead, for a given $\lambda$. If the limit exists then it shouldn't depend on $\lambda$.
The limit on the rays boils down to evaluating:
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3251287",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Let $\{a_n\}$ be a sequence of positive real numbers such that $\sum_{n=1}^\infty a_n$ is divergent. Which of the following series are convergent? Let $\{a_n\}$ be a sequence of positive real numbers such that
$\sum_{n=1}^\infty a_n$ is
divergent. Which of the following series are convergent?
a.$\sum_{n=1}^\infty \frac... | If $\sum_{n=1}^{\infty} a_n$ be a divergent series of positive real numbers prove that the
series $\sum_{n=1}^{\infty}\frac{a_n}{1+a_n}$ is divergent.
Proof:
Let $S_n = a_1 +a_2 + ... +a_n$ .
Since the series $\sum_{n=1}^{\infty} a_n$ is a divergent series of positive real numbers, the
sequence $\{S_n\}$ is a monotone ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3251425",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
Describing the orbits of an algebraic action on $\mathbb{A}^4$ I have an algebraic action of $(k^*)^2$ (an algebraic group) on the variety $\mathbb{A}^4$ by: $(t,u) \cdot (x,y,z,v) = (tx, uy, t^{-1}z, u^{-1}v)$. What are the orbits of this action?
I am not sure exactly how I am supposed to describe these orbits. For ex... | Yes, it is just a big list.
*
*If none of $x,y,z,v$ are zero, then the orbit is a copy of $\mathbb{G}_m^2$: the intersection of two quadrics $x_1x_3=xz$ and $x_2x_4=yv$ in $\mathbb{A}^4$.
*If $x=0$ but $y,z,v$ are nonzero, then you also have a copy of $\mathbb{G}_m^2$ as the orbit: $x_1=0, x_3\neq 0, x_2x_4=yv$. S... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3251992",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Inverse of a function intersecting at y =X line
The inverse of a function intersects the function on $y=x$ line.
This is what I was taught. It works fine for $y=x^2, x^3$ ,
Eg $y = x^2$ meet $x= y^2$ at$ (1,1)$ but..
For a function like $ y =-x^3$
It seems to intersect at $ x+y = 0 , $
Why, is the first statement wr... | Consider the curve $y=1-x$. It's inverse is $y=1-x$, i.e. it is self inverse. This means it intersects all along its curve, despite only intersecting $y=x$ once.
Now suppose a curve $y=f(x)$ intersects the line $y=x$ at $x_0$. This means that $$y_0=f(x_0)=x_0.$$
Applying $f$ to both sides yields
$$
f(y_0)=f(x_0)=x_0,
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3252105",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Dunford decomposition proof : why such a form? I'm working on the proof of the Dunford decomposition theorem :
All matrices $A\in M_n(K)$ such their characteristic polynomials split can be written in the form $A=D+N$ where $D$ is diagonalizable and $N$ is nilpotent.
The proof :
Let $A \in M_n(K) $ a linear operato... | It has this form because each $N_i$ is stable under $A$, so the basis vectors of $B_i$ are sent to combinations of vectors of $B_i$ : their components on $B_j, j\neq i$ are therefore $0$.
More generally, when $E$ is a vector space, $f$ an endomorphism and $F,W$ two stable subspaces such that $E=F\oplus W$, then the ma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3252313",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Given $f(x)=ax^3-ax^2+bx+4$ Find the Value of $a+b$ Let $f(x)=ax^3-ax^2+bx+4$. If $f(x)$ divided by $x^2+1$ then the remainder is $0$. If $f(x)$ divided by $x-4$ then the remainder is $51$. What is the value of $a+b$?
From the problem I know that $f(4)=51$.
Using long division, I found that remainder of $\frac{ax^3-ax^... | $$x=\pm i$$
so
$$a(\pm i)^3-a(\pm i)^2+b(\pm i)+4=0$$
so
$$ai+a\pm bi+4=0$$
$$ai+a+bi+4=0\tag 1$$
or
$$ai+a-bi+4=0\tag 2$$
now we will solve the first equation
$$a+4=0\rightarrow a=-4$$
$$a-b=0\rightarrow b=-4$$
hence $$f(x)=-4x^3+4x^2-4x+4$$
at $x=4$
$$f(4)=-204=-4(51)$$
the second equation gives
$$a=-4$$
$$b=4$$
hen... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3252433",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
} |
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