Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Can the norm of a vector be $\infty$? I am reading Pugh's Analysis and he defines a norm as a certain type of function from $V \to \mathbb{R}$. However, if we have two normed vector spaces, he later says that we can define the operator norm of a linear transformation by
$$||T|| = \sup \left \{\dfrac {|T(v)|}{|v|}: v \... | Calling $\|\cdot\|$ a norm is a small abuse of terminology. Norms must be finite by definition. However, any function satisfying all the norm axioms except for finiteness becomes a norm when restricted to the domain for which it is finite. Even though $\|\cdot\|$ can be infinite, it is a norm on the space of continuous... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3040004",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
What is the Range of $5|\sin x|+12|\cos x|$ What is the Range of $5|\sin x|+12|\cos x|$ ?
I entered the value in desmos.com and getting the range as $[5,13]$.
Using $\sqrt{5^2+12^2} =13$, i am able to get maximum value but not able to find the minimum.
| Another possible approach.
For the first quadrant: $5\sin(x) + 12\cos(x) = 13\sin(x + \arccos(\frac{5}{13}))$. Can follow from there for the rest of the quadrants. Can also try alternative forms for arguments in order to adapt to the values of $x$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3040110",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Invertible matrix properties of a matrix I have here the following question:
Let $X$ be the $5 \times 5$ matrix "full of ones":
$X = \begin{pmatrix}1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1\end{pmatrix}$
$(a)$ Is $X$ invertible? Explain.
$(b)$ Find a number $c... | Hint. You have $M(X+(a-c)I_5)=-a(a-c)I_5$. Pre-multiply by $M^{-1}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3040265",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Using binomial Theorem how we can show $\frac{(x+y)!}{x!y!}\leq \frac{(x+y)^{x+y}}{x^xy^y}$? Using binomial Theorem prove that $$\frac{(x+y)!}{x!y!}\leq \frac{(x+y)^{x+y}}{x^xy^y}.$$
I tried it as follows:
It is clear that $x\leq x+y, \forall x,y\in \mathbb{N}$. Thus, by Binomial Theorem, we have
\begin{align*}(x+y)^{... | $$(x+y)^{x+y}=\sum_{k=0}^{x+y}\frac{(x+y)!}{k!(x+y-k)!}x^{x+y-k}y^k\ge\underbrace{\frac{(x+y)!}{y!(x+y-y)!}x^{x+y-y}y^y}_{\text{evaluated at}\,k=y}=\frac{(x+y)!}{x!y!}x^xy^y$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3040349",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Limit $\lim_{(x, y) \to (\infty, \infty)} \frac{x+\sqrt{y}}{x^2+y}$ Show whether the limit exists and find it, or prove that it does not.
$$\lim_{(x, y) \to(\infty,\infty)}\frac{x+\sqrt{y}}{x^2+y}$$
WolframAlpha shows that limit does not exist, however, I do fail to conclude so.
$$\lim_{(x, y) \to(\infty,\infty)}\frac... | It is enough to observe that, if $y\geq 0$,
$$
x^2 + y \geq \frac{1}{2} (|x| +\sqrt{y})^2,
$$
so that
$$
\left|\frac{x+\sqrt{y}}{x^2+y}\right|
\leq\frac{|x|+\sqrt{y}}{x^2+y}\leq \frac{2}{|x|+\sqrt{y}}.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3040482",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
Permutation probability Let $S = \{1,2,3,4,5,...,n\}$.
Let $\Omega$ be set of permutation maps of $S$.
Let $\Phi : \mathbb{R} \to \mathbb{R}$ be strictly positive and strictly increasing map.
Consider positive function $P: \Omega \to \mathbb{R}$ defined by
$$P(\tau) = \prod_{j=1}^{n} \frac{\Phi(\tau(j))}{\sum_{k=j}^n \... | I do not need the monotonicity of $\Phi$, only positivity. And since we only need the values of $\Phi$ on $\Bbb N_1$, I will assume $\Phi:\Bbb{N}_1\to (0,\infty)$.
We prove this by induction on $n$. And since dealing with many values of $n$ at the same time, things can get confusing. So, I denote $S_n$ and $\Omega... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3041567",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Sum of two co-prime integers I need some help in a proof:
Prove that for any integer $n>6$ can be written as a sum of two co-prime integers $a,b$ s.t. $\gcd(a,b)=1$.
I tried to go around with "Dirichlet's theorem on arithmetic progressions" but didn't had any luck to come to an actual proof.
I mainly used arithmetic pr... | Here's another route you can take to solve this problem. For any $n \ge 7$, you want to show that there is a number $a$ where
*
*$gcd(a, n - a) = 1$,
*$1 < a < n$, and
*$1 < n - a < n$.
One option would be to choose $a$ to be the smallest prime number that doesn't divide $n$. In that case, $gcd(a, n - a) = 1$ be... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3041656",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 2
} |
Minimizer of square root operator norm
Let $A:D(A) \to \mathcal H$ be a positive self-adjoint operator and $\sqrt{A}$ defined by via the spectral theorem on $D(\sqrt{A}) = Q(A)$ where $Q(A)$ is the quadratic form domain.
Let $$E=\inf\{\lVert\sqrt{A}u\rVert^2 : u \in D(\sqrt{A}), \lVert u \rVert = 1\}.$$
Assume th... | Let $A=\int_0^{\infty} \lambda dP(\lambda)$ be the spectral decomposition of $A$. Then $u\in\mathcal{D}(\sqrt{A})$ iff
$$
\|\sqrt{A}u\|^2= \int_{0}^{\infty}\lambda d\|P(\lambda)u\|^2 < \infty.
$$
Suppose $\lambda_0 = \inf \{ \|\sqrt{\lambda}u\| : u\in\mathcal{D}(\sqrt{A}),\;\; \|u\|=1 \}$. If $u_0$ is a minimizer... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3041790",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Winding number of $f$ about an arbitrary point $w_0$ The argument principle tells us given some assumptions of the function $f$ and the contour $\gamma$, we have the winding number of $f$ along $\gamma$ with respect to $0$, $$W(f,\gamma, 0) = Z-P,$$ where $Z$ is the number of the zeros of $f$ and $P$ is the number of p... | The winding number $W(f, \gamma, w_0)$ is defined as
$$ W(f, \gamma, w_0) = \oint_{f \circ \gamma} \frac {1} {w - w_0} dw = \oint_\gamma \frac{f'(z)}{f(z) - w_0 } dz .$$
From this definition, it is clear that
$$ W(f, \gamma, w_0) = W(f - w_0, \gamma , 0).$$
So $W(f, \gamma, w_0)$ counts the number of zeroes minus the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3041869",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
What are the foundations of probability and how are they dependent upon a $\sigma$-field? I am reading Christopher D. Manning's Foundations of Statistical Natural Language Processing which gives an introduction on Probability Theory where it talks about $\sigma$-fields. It says,
The foundations of probability theory... | Probability when there are only finitely many outcomes is a matter of counting. There are $36$ possible results from a roll of two dice and $6$ of them sum to $7$ so the probability of a sum of $7$ is $6/36$. You've measured the size of the set of outcomes that you are interested in.
It's harder to make rigorous sense ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3042059",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
"answer_count": 2,
"answer_id": 0
} |
Counting the directed paths in a particular directed graph I want to find out how many directed simple paths from $s$ to $t$ are in the following directed graph $G=(V,E)$.
$$\begin{align}
V=&\{s, v_1, v_2,\ldots, v_n, t\}, \quad n=2k, k \in \mathbb{N} \\
E=&\{ (s, v_1), (s, v_2), \\
&\;(v_1,v_3), (v_1,v_4), (v_2,v_3),(... | For each vertex of the graph, the level of the vertex is the length of the shortest directed path from $s$ to that vertex. So $s$ has level $0$, $v_{2l-1}$ & $v_{2l}$ have level $l$, and $t$ has level $k+1$. Note that each directed path from $s$ to $t$ much contain exactly one vertex from each level.
Let $s=u_0 \to u... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3042392",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Simple two variable am-gm inequality Given $x,y \in \Bbb{R}$, show that:$$x^2+y^2+1\ge xy+y+x $$
I tried using the fact that $x^2+y^2 \ge 2xy$ But then I'm not sure how to go on, Also tried factoring but didn't help much, also tried substituting $\frac{x^2+y^2}{2}$ instead of $xy$ but that gave me the same result of th... | We need to prove that
$$y^2-(x+1)y+x^2-x+1\geq0,$$ for which it's enough to prove that
$$(x+1)^2-4(x^2-x+1)\leq0$$ or $$(x-1)^2\geq0.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3042470",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Simplification of $ \sqrt{(1-x^2)}$ to $(1-\frac{x^2}{2})$ While following a proof from an electrical engineering book (Design of Analog CMOS Integrated Circuits, second edition from Behzad Razavi ), I came across a simplification which I found curious. In equations 14.18 to 14.19 they state that the following holds fo... | Term to look for: linear approximation
In general, the best linear approximation for a differentiable function near a point $c$ is
$$
f(x) \approx f(c) + f'(c)\;(x-c)
$$
This is essentially the definition of the derivative. And you should find this in your calculus book soon after the definition of derivative.
Now ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3042624",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Why is $\omega^{\omega}$ countable? I'm confused as to why $|\omega^{\omega}| \neq \aleph_0^{\aleph_0}$. Since
\begin{align}
\omega^{\omega} = \left\lbrace \sum_{i < \omega}^{1} (\omega^i \cdot n_i) + n_0 : n_i,n_0 \in \mathbb{N}_0 \right\rbrace
\end{align}
wouldn't it follow that $|\omega^{\omega}|$ can be represented... | No, $\omega^\omega$ is the set of those which can be represented as finite sequences. Namely, an ordinal below $\omega^\omega$ is a polynomial in $\omega$. So in effect $\omega^\omega$ is the natural way to represent $\Bbb N[x]$, which is of course countable.
So it does not correspond to the branches in a tree of heigh... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3042755",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Calculate the limit $\lim_{x\rightarrow 0}\frac{x^2 \cos\left(\frac{1}{x}\right)}{\sin(x)}$. We could use L'Hospital here, because both numerator as well as denominator tend towards 0, I guess. The derivative of the numerator is $$x^2\cdot \left(-\sin\left(\frac{1}{x}\right)\right) \cdot \left( -\frac{1}{x^2}\right) + ... | Hint: Without using L’Hôpital’s Rule, note that
$$\frac{x^2\cos\big(\frac{1}{x}\big)}{\sin x} = \frac{x}{\sin x}\cdot x\cos\bigg(\frac{1}{x}\bigg)$$
and recall $\lim_\limits{x \to 0}\frac{\sin x}{x} = 1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3042924",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Help solving a digit and word code problem/riddle: sum of four "ONE"s is "TEN"
Replace each letter by a digit. The same digit must represent each
letter, and no beginning letter of a word can be zero. No two letters can
be the same number. Find the digits represented by the letters 'O',
'N', 'E', 'T'.
O N E
O N E
O N ... | 'ONE' represents the number $100 \times O + 10 \times N + E$, just as $781$ represents $7 \times 100 + 8 \times 10 + 1$. This is what the decimal position
system is.
So 'ONE' added 4 times to itself is just $400 \times O + 40 \times N + 4E$ and this should represent the same number as 'TEN' = $100 \times T + 10\times ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3042979",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Operator norm $ ( \ell_2 \to \ell_1)$ Let $X$ be a finite dimensional normed vector space and $Y$ an arbitrary normed vector space. $ T:X→Y$.
I want to calculate $\|T\|$ for where $X = K^n$, equipped with the Euclidean norm $\|\cdot\|_2$, $Y := \ell_1(\mathbb{N})$ and $Tx := (x_1,\ldots,x_n,0,0,\ldots) \in \ell_1(\math... | I will elaborate on my comment above
Given an operator $T: X \rightarrow Y$ where $$X = \mathbb{K}^{n}$$
$$Y = l^{1}(\mathbb{N})$$ its norm is given by
$$||T||_{op} = \sup_{x \neq 0}{\frac{||Tx||_{1}}{||x||_{2}}} = \sup_{||x||_{2} \leq 1}{||Tx||_{1}} = \sup_{||x||_{2} = 1}{||Tx||_{1}}$$
So we have to maximize $$||Tx||_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3043100",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Operator norm on Lebesgue integrable functions Let $L_1([0,1],m)$ be the Banach space of $\mathbb{K}$-valued integrable functions with respect to Lebesgue measure $m$, where $\mathbb{K}$ is either $\mathbb{R}$ or $\mathbb{C}$. The norm on this space is defined like this: $||f||_1=\int_{[0,1]}|f| \ dm$. I have to show t... | Answer to part a): it is not necessary to get equality in $|\int_{[0,1]} f(x)n\sin(n^{2}x)\,dx | \leq \|f\|_1\|g\|_{\infty}$. Instead, we get an 'approximate equality' as follows: let $\epsilon >0$. Choose $\delta >0$ such that $\sin\, x>1-\epsilon$ for $\frac {\pi} 2 -\delta <x <\frac {\pi} 2 +\delta $. Let $f=\frac ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3043231",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Verifying that $ \prod_{j=1}^{\infty} \frac{1}{1-q^j} = \prod_{j=1}^{\infty} \frac{1}{(1-q^{2j-1})(1-q^{2j})}$ On page 165 of Chapter 13, how was the equality made from line 1 to line 2?
https://archive.org/details/NumberTheory_862/page/n173
Namely, how $$ \prod_{j=1}^{\infty} \frac{1}{1-q^j} = \prod_{j=1}^{\infty} \fr... | Simply pair-off factors and use the fact that multiplication is commutative:
\begin{align*}
\prod_{j=1}^{\infty}\frac{1}{(1-q^{j})} & = \frac{1}{1-q}\frac{1}{1-q^2}\frac{1}{1-q^3}\frac{1}{1-q^4}\cdots\\
& = \left(\frac{1}{1-q}\frac{1}{1-q^2}\right)\left(\frac{1}{1-q^3}\frac{1}{1-q^4}\right)\cdots\\
& = \left(\frac{1}{1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3043390",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Analytical approximation for logit-normal-binomial distribution As I understand, there is no closed form expression for
$$f(x, \mu, \sigma) = \int_0^1 p^{(x-1)}(1-p)^{n-x-1}\exp\left(-{(\text{logit}(p) -\mu)^2 \over 2\sigma^2}\right)dp.$$
Is it possible to obtain an analytical approximation for this?
| Here's what you need to do:
*
*Decide on an interpolator. I suggest a tricubic b-spline, but finding software for this is going to be painful. To understand this interpolant, start in Rainer Kress's Numerical Analysis which introduces it in 1D, learn about the bicubic b-splines in 2D, and then you'll be able to unde... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3043544",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Limit Question $\lim_{x\to\infty} \sqrt{x^2+1}-x+1$ I understand the answer is 1 which kind of makes sense intuitively but I can't seem to get there. I would appreciate if someone pointed out which line of my reasoning is wrong, thanks. I tried writing all my steps
\begin{equation}
\lim_{x\to\infty} \sqrt{x^2+1}-x+1... | From here we have
$$\frac{\left( \sqrt{x^2+1}-(x-1) \right) \left( \sqrt{x^2+1}+(x-1) \right)}{\sqrt{x^2+1}+(x-1)}=\frac{(\sqrt{x^2+1})^2-(x-1)^2}{\sqrt{x^2+1}+(x-1)}=$$$$=\frac{x^2+1-x^2+2x-1}{\sqrt{x^2+1}+(x-1)}=\frac{2x}{\sqrt{x^2+1}+(x-1)}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3043648",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Find locus of points by finding eigenvalues
Let $\boldsymbol{x}=\left(\begin{matrix}x\\ y\end{matrix}\right)$ be a vector in two-dimensional real space. By finding the eigenvalues and eigenvectors of $\boldsymbol{M}$, sketch the locus of points $\boldsymbol{x}$ that satisfy $$ \boldsymbol{x^TMx}=4$$
given that
$$... | The eigenvectors are orthogonal and span $\Bbb R^2$. This means $\mathbf v=\begin{bmatrix}x\\y\end{bmatrix}=c_1\mathbf x_1+c_2\mathbf x_2$.
$\mathbf v^TM\mathbf v=(c_1\mathbf x_1^T+c_2\mathbf x_2^T)M(c_1\mathbf x_1+c_2\mathbf x_2)\\=(c_1\mathbf x_1^T+c_2\mathbf x_2^T)(c_1\lambda_1\mathbf x_1+c_2\lambda_2\mathbf x_2)\\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3043790",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Reason for the integer case and the rational case to be solved differently
Assume $f$ is continuous,$f(0)=1$ , and $f(m+n+1)=f(m)+f(n)$
for all real $m, n$.
Show that $f(x) = 1 + x$ for all real numbers $x$.
This is referenced from Terence Tao’s solving mathematical problems and in the exercise he provided a hin... | Before I proceed with my solution let me tell you an interesting Theorem which is necessary to understand the solution I have given.
Continous Additive functions are linear
Coming back to your Orignal Problem
Consider the function $g(x)=f(x)-1$. Note that $g(x)$ is continuous function.
Keep it aside for a while.
By the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3044092",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Uniformly bounding a sequence $(\mathbf\Lambda_n^{-1})_{n=1}^\infty$ of inverses of bounded linear operators Suppose we wish to prove the following.
Let $X$ be a Banach space and let $(\mathbf\Lambda_n)_{n=1}^\infty \subset \mathcal L(X)$ be a sequence of invertible bounded linear operators on $X$ that converges in op... | Hints, in easier-to-type notation: First, it's enough to consider the case $T_n\to I$, where $I$ is the identity, because... . And for that case, note that if $||T||<1$ then $$(I-T)^{-1}=I+T+T^2+\dots.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3044196",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Suggestion for a functional equation $f'\left(\frac{a}{x}\right)=\frac{x}{f(x)}$ where $f:(0,\infty)\to(0,\infty)$ is differentiable I want to solve the following functional equation:
Find all differentiable functions $f : (0,\infty) \rightarrow (0,\infty)$ for which there is a positive real number $a$ such that $$f'\... | It seems that $f \approx x^\beta $ might work.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3044362",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
when does $\int_{0}^{\infty}\frac{\sin(t)}{(t+1)^\alpha}dt$ converge? The original question was to determine for which values of $\alpha \in \mathbb R$ does the integral $$\iint_{\mathbb R^2}\frac{\sin(x^2+y^2)}{(x^2+y^2+1)^\alpha}dxdy$$ converge.
I managed to simplify this and even reach a partial answer:
$$\iint_{\ma... | Hint (for the main question, not the original inspiring question): Try integration by parts with $u = (1+t)^{-\alpha}$, $dv = \sin t\,dt$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3044482",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Calculate $\int_{-\infty}^\infty \frac{e^{-x^2}}{x^2+a^2}\ dx$. Let $$F(a)=\int_{-\infty}^\infty \frac{e^{-x^2}}{x^2+a^2}\ dx, \quad a>0.$$ Is it possible to relate $F(a)$ to some known (special) functions?
| Parameterize this integral by adding a second parameter, $t$:
$$I(t):= \int_{-\infty}^\infty \frac{e^{-(x^2+a^2)t}}{x^2+a^2}dx$$
Differentiating with respect to $t$, we have
$$I'(t)=-\int_{-\infty}^\infty e^{-(x^2+a^2)t}dx=-\sqrt{\frac{\pi}{t}} e^{-a^2 t}$$
This shows us that
$$\begin{align}
I(t)
&=I(0)-\int_0^t \sqrt{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3044618",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Constrained Optimization Geometry Confusion In a constrained optimization problem, let's consider the example $$\begin{cases}f(x,\ y) = yx^2\ \Tiny(function\ to\ be\ maximized) \\ g(x,\ y) = x^2 + y^2 = 1\ \Tiny(constraint)\end{cases}$$ why does the answer not need to satisfy $f(x^*,\ y^*) = 1$? Geometrically, viewing... |
Geometrically, viewing $f(x, y)=yx^2$ and $g(x, y)=x^2+y^2$ in $R^3$ (which motivated this question), why aren't solutions required to be points where $f(x, y)$ and $g(x, y)$ intersect, or at least where $f(x, y)$ intersects $g(x, y)=1$?
You are right, $g(x,y)=x^2+y^2$ is a two-variable function, whose graph is para... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3044697",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How many sequences can be made with 5 digits so that the difference between any two consecutive digits is $1$? Using the digits $0$, $1$, $2$, $3$, and $4$, how many ten-digit sequences can be written so that the difference between any two consecutive digits is $1$?
I was wondering if my solution is right.
Let $a(n)$ ... |
here is your answer. Your approach was absolutely correct.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3044871",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 1
} |
Counterexample PID We know that if F is a field, then the polynomial ring over F is a PID.
Do you have a counterexample that shows that if F isn’t a field than the polynomial ring over F isn’t a PID?
| You can even show, extending slightly the argument in the other answer, the following.
Suppose $A$ is a domain. Then $A[x]$ is a PID iff $A$ is a field.
If $A$ is a field, then $A[x]$ is Euclidean, and thus a PID.
Suppose now $A[x]$ is a PID. Let $0 \ne a \in A$. We want to show that $a$ is a unit in $A$.
Consider th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3045034",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Proving that $B(X,Y) $ is a Banach Space if $Y$ is. Let $B(X,Y)$ be the family of all bounded maps from $X$ to $Y,$ normed linear maps. Then, $B(X,Y) $ is a Banach Space if $Y$ is.
Remark: I've seen this question before $Y$ is a Banach space if $B(X,Y)$ is a Banach space, but it is the converse of my question statement... | Credits to Olof Rubin. So, I post the full proof for future readers.
Let $\{T_n\}_{n=1}^{\infty}\in B(X,Y)$, be a Cauchy sequence and $\epsilon>0$ be given. Then, there exists $N$ s.t. forall $m\geq n\geq N,$
$$ \|T_n-T_m\|<\epsilon.$$
Since $$ \|T_n-T_m\|=\sup\limits_{\|x\|\leq 1}\|T_n(x)-T_m(x)\|,\;\;\forall\;m,n\in ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3045155",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
1-norm and symmetry Define the fidelity function for positive operators by $F(\rho, \sigma) = \lVert \sqrt{\rho}\sqrt{\sigma}\rVert_1$. Here, $\lVert\cdot\rVert_1$ is the Schatten 1-norm and defined as $\lVert A\rVert_1 = \operatorname{Tr}(\sqrt{A^{\dagger}A})$.
I'm having some trouble showing that $F$ is symmetric in ... | We know that $F( \rho, \sigma) = \text{Tr}( \sqrt{ \sqrt{\rho} \sigma \sqrt{\rho}})$. Now consider
\begin{align*}
F( \sigma, \rho) &= \lVert \sqrt{\sigma} \sqrt{\rho} \rVert_1 = \text{Tr} \big( \sqrt{(\sqrt{\sigma} \sqrt{\rho})^\dagger \sqrt{\sigma} \sqrt{\rho}} \big) \\
&= \text{Tr}\big( \sqrt{ \sqrt{\rho} \sigma \sqr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3045456",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Maximal ideal in ring of power series If $R$ is a commutative ring with identity we know that the maximal ideals of the ring of power series over $R$ have the form $M’=(M,x)$ where $M$ is a maximal ideal of $R$. Do you have a counterexample that shows that if $R$ doesn’t have an identity then the theorem doesn’t hold?
... | I don't know for sure if this suits your needs or not, but if $R=2\mathbb Z/4\mathbb Z$ and $M=(2x)\lhd R[[x]]$, then $R[[x]]/M\cong R$ has two elements, so $M$ is maximal (in the sense you specified in the comments.)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3045623",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Solve the system of equations in the set of real numbers. Solve the system of equations in the set of real numbers:
$$\begin{cases}
\frac1x + \frac1{y+z} = \frac13 \\
\frac1y + \frac1{x+z} = \frac15 \\
\frac1z + \frac1{x+y} = \frac17
\end{cases}$$
I got:
$$\begin{cases}
3(x+y+z)=x(y+z) \\
5(x+y+z)=y(x+z) \\
7(x+y+z)=... | We know that via your equations, $$\frac{15}2(x+y+z)=xy+yz+xz$$Hence, $$xy=\frac12(x+y+z)$$$$yz=\frac92(x+y+z)$$$$xz=\frac52(x+y+z)$$So, assuming $x+y+z\neq0$, $z=9x$, $z=5y$. Try using this to move forward!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3045794",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
What is the meaning of $\mathbb{Z}_{5}^{+}$ (and $\mathbb{Z}_{5}^{*}$) in group theory? What does $\mathbb{Z}_{5}^{+}$ mean? I know $\mathbb{Z}_{5}$ represents the set of integers modulo 5. I would assume this would mean it is the set of integers modulo 5 under addition except that normally this is notated as $(\mathbb... | The notation $\mathbb{Z}_n^{\times}$ is sometimes used to denote the group of units modulo $n$ with respect to multiplication, so I would presume that $\mathbb{Z}_5^+$ denotes the additive group of integers modulo $5$, in order to contrast it with the multiplicative group.
[For what it's worth, I'll point out that $\ma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3045915",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Finiteness of the normalization of an algebra over a DVR Let $R$ be a DVR, $A$ a finitely generated integral $R$-algebra, and $A'$ the normalization of $A$ in the fraction field of $A$.
Then is $A'$ finite as an $A$-module?
I know that if $R$ is a field, then it's true.
And I know this is true, see here, section 2 in p... | The property you state is related to the property of $R$ being excellent, and more specifically to weaker condition of whether $R$ has the N-2 property or the stronger Nagata property; see [Matsumura, §31] and [Illusie–Laszlo–Orgogozo, Exp. I].
Not all DVR's satisfy your property. We give an example below, due to Nagat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3046024",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Proving the following quadratic inequality? Apologies if this has been asked before - I could not find a question with this exact inequality.
Basically the inequality is
$$(a+b+c)^2 \leq 3 a^2 + 3 b^2 + 3 c^2$$
Expanding it out we see that
$$(a+b+c)^2 = a^2 +b^2 + c^2 + 2ab + 2bc + 2ac$$
so I guess it is equivalent to ... | It follows immediately from Cauchy-Schwarz:
$$(a+b+c)^2 = (1\cdot a + 1 \cdot b + 1 \cdot c)^2\leq (1^2+1^2+1^2) (a^2 + b^2 + c^2)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3046157",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 0
} |
A basic power problem.
What is unit digit of $14^{15^{16^{17}}}$? (no brackets given)
This question is simple and it is based on cyclicity, but the confusion arise when I taken this approach:
We know that $5$ raised to any natural number will give $5$ in the unit place. Which will make the number odd.
And the cyclici... |
So basically to cut the long story short how do we resolve a^b^c ? Do we find a^b and the the answer of that is raised to c? Or first we resolve b^c and the answer is raised to a?
Most people who have an opinion uses $b^c$ first, then raise $a$ to the result. Calculating $a^b$, then raise that to the power of $c$ is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3046263",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Geometry problem (from national competition) Given right angled triangle $\triangle ABC$ with right angle at point C.
Let points $D, E$ lie on $AB$, such that $|BC|=|BD|$ and $|AC|=|AE|$.
Let point $F$ be orthogonal projection of point $D$ onto $AC$ and let point $G$ be orthogonal projection of point $E$ onto $BC$.... | Drop the altitude from $C$ to $AB$ which cuts $AB$ at $X$. Then it is enough to prove that $DX=DF$. Say $\angle ABC = 2x$, then $$ \angle BDC = \angle DCB = 90-x$$ and so $\angle DCX = x$. Clearly we have $$\angle ACX = 90-\angle XCB = 2x,$$ so $\angle FCD = x$.
So triangles $FCD$ and $XDC$ are congurent (a.s.a.) and t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3046560",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How does the image of the Hurewicz map $\pi_n(X,x) \to H_n(X)$ depend upon the choice of the base point? Let $X$ be a path connected topological space. I understand that the homotopy groups $\pi_n(X,x_0)$ and $\pi_n(X,x_1)$ are isomorphic to each other. However I do not understand whether the image of the Hurewicz map ... | Note that we don't just have some arbitrary isomorphism $\pi_n(X,x_1)\to \pi_n(X,x_0)$; we have an explicit description of what the map is. Namely, we can get such an isomorphism by picking a path $\gamma$ from $x_0$ to $x_1$ and then inserting copies of $\gamma$ radially starting at the basepoint $s_0$ of $S^n$ to tu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3046711",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Proof that $ \sum_{n=2}^{\infty} \frac{2}{3^n \cdot (n^3-n)} = \frac{-1}{2} + \frac{4}{3}\sum_{n=1}^{\infty} \frac{1}{n \cdot 3^n}$ Task
Proof that $ \sum_{n=2}^{\infty} \frac{2}{3^n \cdot (n^3-n)} = -\frac{1}{2} + \frac{4}{3}\sum_{n=1}^{\infty} \frac{1}{n \cdot 3^n}$
About
Hi, I have been trying to solve this task sin... | hint
$$\frac{2}{n^3-n}=\frac{-2}{n}+\frac{1}{n-1}+\frac{1}{n+1}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3046791",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Apostol's Calulus: Prove that $[x+y] = [x]+[y]$ or $[x]+[y]+1$, where $[·]$ is the floor function.
Prove that $[x+y] = [x]+[y]$ or $[x]+[y]+1$, where $[·]$ is the floor
function
I'm Having a little bit of trouble with the last part of this proof.
First, I will use the definition of floor function:
$[x] = m ≡ m ≤ x ... | Alternatively. By definition $[x+y]$ is the largest possible integer that this less than or equal to $x+y$. But $[x] \le x$ and $[y] \le y$ so $[x] + [y] \le x+y$. So $[x]+[y] \le [x+y]$.
Likewise $[x+y] + 1$ by definition is the smallest possible integer that is larger $x + y$. But $[x]+ 1 > x$ and $[y] + 1 > y$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3046926",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 2
} |
When will a road reach a certain condition? I have an exponential regression equation that is designed to predict the future condition of a road at a certain age:
condition = 21-EXP(0.06*age)
Note: Road condition is a range from 1 to 20; a road with a condition of 20 is in perfect condition.
Question:
I know that a ro... | Hint #1: Call the equation $$C = 21 - e^{0.06A}$$ How would you solve this in terms of A?
Hint #2: When would the road be in the worst condition? How many years would this take?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3047042",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
using Bayes’ Rule to calculate conditional probability i have following problem,
"Students who party before an exam are twice as likely to fail the exam as those who don't party (and presumably study). Of 20% of students partied before the exam, what percentage of students who failed the exam went partying?"
i believe ... | Let $x$ be the total number of students and $p$ be the probability of a student who didn't party failing the exam. The probability of a student who partied before the exam failing the exam is then $2p$.
$x/5$ students partied, out of which $2px/5$ failed. Out of the $4x/5$ who didn't party, $4px/5$ failed the exam. The... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3047159",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Does there exist any probability density function $f:\mathbb{R}\to\mathbb{R}$ which is not Riemann integrable? Let $f:\mathbb{R}\to\mathbb{R}$ be a probability density function. Can the following be happened for $f$?
(1) $f$ is not integrable on an (some) interval of $\mathbb{R}$.
(... | It is known that there exists a measurable set $E$ in $\mathbb R$ such that $0<m(E\cap I) <m(I)$ for every open interval $I$. If $f=\frac {I_E} {m(E)}$ then $f$ is a density function but it is not continuous at any point so it is not Riemann integrable on any interval.
For the construction of such a set $E$ see Creati... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3047260",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Factorising $99999\,00000\,99999\,00001$ I am reading an article in The Mathematical Gazette about factorising the 20 digit number
$$N=99999\,00000\,99999\,00001.$$
It is stated that $N=\dfrac{10^{25}+1}{10^5+1}$ (I understand why this is true), and consequently, if $p$ is a prime factor of $N$, then $p$ must be of the... | If $p$ is a prime factor of $N$, then it is a factor of $10^{25}+1$ and thus $$10^{25}\equiv -1 \implies 10^{50}\equiv 1\pmod p,$$
i.e. if $d$ is the order of $10$ modulo $p$, then $d$ divides $50$. But $d$ cannot divide $25$ (otherwise we would have $10^{25}\equiv 1\pmod p$), and $d$ cannot divide $10$ either (otherwi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3047384",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
fix point solution or approximation available? logistic regression? please, is there a simple closed-form or approximation to the following fixed-point problem in $x$? $x$ is the value searched for. $m$, $g$ and $N$ are real parameters, all greater than 0,
\begin{equation}
x=\frac{1}{1+m g^{-N x}}
\end{equation}
any i... | I am afraid that there is no closed form of the solution.
Without any information about $x$, let us consider the function
$$f(x)=x \left(1+m g^{-n x}\right)-1$$ What we have is $f(0)=-1$ and $f(1)=m g^{-n} >0$.
We also have to notice that
$$f(0)=-1 \qquad f'(0)=1+m \qquad f''(0)=-2\, m\, n \log (g)$$ So, if $g>1$, $f(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3047439",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Drawing balls with a finite number of replacement I have to solve this problem:
"Suppose a box contains $18$ balls numbered $1–6$, three balls with each number. When $4$ balls are drawn without replacement, how many outcomes are possible?". (The order does not matter).
I can't find a simple formula for it.
I've tried i... | One more way is to use a generating function. Consider
$$F(x) = (1+x+x^2+x^3)(1+x+x^2+x^3)(1+x+x^2+x^3)\cdots(1+x+x^2+x^3) = (1+x+x^2+x^3)^6$$
Looking at the first $(1 + x+ x^2 + x^3)$ term, we can think of the exponent of $x$ as representing the number of "1" balls we choose (i.e. $1=x^0 \rightarrow 0,\ x = x^1\righta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3047584",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Sequence problem regarding convergence from an online contest Let $(x_n)_{n\in \mathbb{N}}$ be a sequence defined by $x_0=1$ and $x_n=x_{n-1}\cdot \big(1-\frac{1}{4n^2}\big)$, $\forall n\geq 1$.
Prove that:
a) $(x_n)_{n\in \mathbb{N}}$ is convergent
b) if $l=\lim_{n\to \infty} x_n$, compute $\lim_{n \to \infty} (\frac{... | Partial Answer
We have $$x_n=\prod_{k=1}^{n} \big(1-\frac{1}{4k^2}\big){=\prod_{k=1}^n{(2k-1)(2k+1)\over (2k)^2}\\=\prod_{k=1}^n{(2k-1)\cdot 2k\cdot 2k\cdot(2k+1)\over (2k)^4}\\={1\over 16^n}\cdot {1\over(n!)^4}\prod_{k=1}^n(2k-1)\cdot 2k\cdot 2k\cdot(2k+1)\\={(2n)!\cdot (2n+1)!\over 16^n\cdot (n!)^4}}$$therefor by usi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3047723",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Properties of length preserving linear transformations Let $T:\mathbb{R}^n\rightarrow\mathbb{R}^n$ be a linear transformation that preserves length. That is $||T(x)||=||x||$ for all $x\in\mathbb{R}^n$.Then:
*
*If $\langle x,y\rangle=0$ then $\langle T(x),T(y)\rangle=0$.
*Show that columns of the matrix of $T$ in t... | If
$\Vert T(z) \Vert = \Vert z \Vert, \; \forall z \in \Bbb R^n, \tag 1$
then
$\langle T(z), T(z) \rangle = \Vert T(z) \Vert^2 = \Vert z \Vert^2 = \langle z, z \rangle, \; \forall z \in \Bbb R^n, \tag 2$
so with
$z = x + y, \tag 3$
$\langle T(x + y), T(x + y) \rangle = \langle x + y, x + y \rangle; \tag 4$
now,
$\langl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3047878",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
The boundary of $r$-neighborhood of $p$ is not equal to sphere of radius $r$ at $p$
(a) Find a metric space in which the boundary of $r$-neighborhood of $p$ is not equal to sphere of radius $r$ at $p$
(b) Need the boundary be contained in the sphere?
My attempt.
(a) I think in a discrete space $M$ with the discrete m... | For any open ball $B(p,r)$, its boundary $\partial B(p,r)$ is always contained in the sphere $S(p,r)$.
Indeed, first notice that the closure $\overline{B(p,r)}$ of the open ball is always contained in the closed ball $\overline{B}(p,r)$. This follows from the fact that $\overline{B}(p,r)$ is a closed set.
For any set $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3047979",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Determinant of a $n\times n$ matrix in terms of $n-1\times n-1$ matrix Suppose we know the determinant of matrix $A=(a_{ij})_{i,j=1}^{n-1}$. Can we express determinant of matrix $A'=(a_{ij})_{i,j=1}^{n}$ in terms of the determinant of matrix $A$?
We see that the matrix $A'$ differs from matrix $A$ only in the extra ri... | No, e.g., consider:
$$\begin{array}{ccc}
A = \left(\matrix{0 & 0 \\ 0 & 1}\right) & \quad\quad & B = \left(\matrix{1 & 0 \\ 0 & 0 }\right) \\
A' = \left(\matrix{0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 1}\right) & \quad\quad & B' = \left(\matrix{1 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 1}\right)
\end{array}$$
$A$ and $B$ have the sa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3048074",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Closed form for recursive sequence mod p When we have recursive sequences, we often seek to define them in a closed form if possible. Yet sometimes, these recursive sequences don't have closed forms. So my question is, is there any recursively defined sequence which doesn't have a closed form, but does have a closed fo... | It is really much harder than you think to nail down precisely what "closed form" means. Your definition in the comments is really not sufficient: for example, does "$a_n = 1$ if $n$ is prime and $0$ otherwise" count as a closed form?
Anyway, assuming you're only asking about one prime, the answer is yes for dumb reaso... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3048160",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Dimension of nullspace and number of rows
A matrix $A$ has $10$ columns and dim(Null($A^{T}$ ))$=7$. The smallest possible number of rows of $A$ is
$(A)$ $5$
$(B)$ $6$
$(C)$ $7$
$(D)$ $8$
$(E)$ $9$
I know that dim(Null($A^{T}$ ))$=7$ implies that there are $7$ rows of zeros and that:
Rank($A$)+Nullity($A^T$) $=$ # of... | Hint: You want Rank($A$)+Nullity($A^T$) to be as low as possible. You already know how large the right term is. What's the lowest possible the left term could theoretically be? What would the resulting matrix be?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3048223",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Asymptotic solution I am looking for asymptotic solutions to the equation
$$\alpha^{-1}x+\sqrt{\pi}\frac{\sqrt{x}}{2}\text{erf}\left(\frac{\sqrt{x}}{2}\right)=\beta^{-1}e^{-x/4},\qquad \alpha\ll1,\beta\gg1.$$ When $\alpha$ is large and the first term is negligible, this is easy to do, but I don't know how to proceed wi... | Let $\tilde \beta = 1/\beta$. Multiplying by $\alpha$ and getting rid of the square roots, we can rewrite the equation as
$$x - \alpha \tilde \beta e^{-x/4} +
\alpha x \int_0^{1/2} e^{-x t^2} d t = 0.$$
Now we can look for $x$ in the form $\sum c_{i,j} \alpha^i \tilde \beta {}^j$ by substituting the sum into the equat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3048369",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Inequality with power and condition I have this to propose :
Let $a,b,c,d$ be real positive numbers such that $abcd=1$ then we have : $$\sum_{cyc}a^{ab}\geq 4$$
First I definitively can't prove this by my own but if someone can prove this it would be very helpful to demonstrate this : Prove that $a^{ab}+b^{bc}+c^{cd... | Here I have proved this
$$n+\sum_{cyc}\ln(a_i^{a_i a_{i+1}})\leq \sum_{cyc}a_i^{a_i a_{i+1}}$$
It's easy to conlude if we note that with $\prod_{i=1}^{n}a_i=1$ we have :
$$\sum_{cyc}\ln(a_i^{a_i a_{i+1}})\geq 0$$
Now put $n=4$ and we have your result .
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3048497",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
About every subgroup of $ ( \mathbb{Z} , + ) $ being cyclic. I'm citing below the definition of and a theorem about cyclic groups, as it is written in my book (Algebra, by Thomas Hungerford):
Definition:
Let $ G $ be a group (notation is multiplicative in here.) For every $ a \in G, $ a cyclic group is: $ \langle a \r... | $ H$ closed under subtraction $\Rightarrow$ $H$ closed under remainder $= \bmod $ (via repeated subtraction) hence $H$ is closed under $\gcd,\,$ since gcds are computable by repeated $\!\bmod$ (or subtraction) by Euclid. Now it's easy to show $H$ is generated by the gcd of its elements - its least positive element (for... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3048767",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
$f$ holomorphic in $\mathbb{D}$. Prove $f$ has a zero in $\mathbb{D}$ Let $f$ be holomorphic in $\mathbb{D}$ and $f$ be continuous on $\overline{\mathbb{D}}$. Assume $f(0)=c$ and $|f(z)|>|c|$ for $|z|=1$. Prove that $f$ has a zero in $\mathbb{D}$.
Since it's dealing with the number of zeros (or existence of), My initia... | Apply the minimum principle: $\lvert f\rvert$ must have a minimum somewhere, but it can't be attained at a $\omega$ such that $\lvert\omega\rvert=1$ (because $\bigl\lvert f(w)\bigr\rvert>\bigl\lvert f(0)\bigr\rvert$. Therefore, it is attained at some $\omega$ with $\lvert\omega\rvert<1$. Therefore, by the minimum princ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3048872",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Does $\text{SO}_2(\mathbb{Q}_5)$ contain non-trivial elements? I was trying to find an element of $\text{SO}_2(\mathbb{Q}_5)$ for the $5$-adic numbers. By analogy with $\text{SO}_2(\mathbb{R})$
$$ \left[ \begin{array}{rr} a & -b \\ b & a \end{array} \right] \text{ with } a^2 + b^2 = 1$$
When we solve this equation mo... | Short answer: Your group is isomorphic to $\Bbb Q_5^\times$. Indeed, there are isomorphisms of topological groups (inverse to each other):
$$SO_2(\Bbb Q_5) \rightarrow \Bbb Q_5^\times$$
$$ \pmatrix{a & -b \\b &a} \mapsto a+ ib$$
and
$$\Bbb Q_5^\times \rightarrow SO_2(\Bbb Q_5)$$
$$x \mapsto \frac12 \pmatrix{x +x^{-1} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3048961",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
Let $P$ be prime and contain $IJ$, the product ideal. Then $I \subset P$ or $J \subset P$ So I already witness the solution. It is this: Assume $I \not\subset P$, then there is $i \not\in P$. Then the product $ij \in IJ \subset P$, but since $P$ is prime, $i \in P$ or $j \in P$, so $J \subset P$.
here is what I don't g... | The proof as written is phrased a little awkwardly and missing important details. Here is a correct proof (with everything spelled out):
Suppose $IJ \subset P$ and $I \subsetneq P$. We wish to show that for all $j \in J$, we have $j \in P$. Fix $j \in J$ $i \in I \setminus P$ and note that $ij \in IJ$. Since $IJ \sub... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3049240",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Stability Systems - Duffing oscillator In the case a=1,b=-1 this is the system:
$$ dx=y $$
$$ dy=-x + x^3$$
I have to draw the phase space with the trajectories of the orbits. And I don´t know who to demonstrate the direction in the orbits. I only know is a circle for the $(0,0)$ and hyperbola for$(-1,0),(1,0)$.
| Form the Jacobian of the system :
$$J(x,y) = \begin{bmatrix} 0 & 1 \\ -1 + 3x^2 & 0\end{bmatrix}$$
For the origin $O(0,0)$ which is a critical point for the given system, it is :
$$J(0,0) = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$$
Then, the eigenvalues of the given Jacobian for the origin :
$$\det(J(0,0) -\lamb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3049293",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Solving Generating Function when there is condition on two variables. " Find the number of ways of giving 10 identical gift boxes to 6 people : A, B, C, D, E, F in such a way that total number of boxes given to A and B together does not exceed 4. "
I tried it in this way :
[$x^{10}$] $(1+x^{1}+...+x^{4})^{2}*( 1+x^... | What you've to find is $x_1+x_2+x_3+x_4+x_5+x_6=10$ (I've replaced A-F by 1-6). Further, $x_i\geq 0$ and $x_1+x_2\leq 4$.
Required answer= $$\sum_{i=0}^{4}(Coeff.\ of\ x^i\ in \ (x^0+x^1+\cdots x^4)\cdot (x^0+x^1+\cdots x^4))\cdot(Coeff.\ of \ x^{10-i}\ in \ (x^0+x^1+\cdots x^{10})^4)$$
Can you solve it now?
In open ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3049420",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Bean machine (Galton board) explanation I am confused by the marked part of the following explanation (see below; https://en.wikipedia.org/wiki/Bean_machine). Suppose the bead bounces to the right twice and to the left also twice. It will land exactly at the center. But according to this explanation it will land to sec... | There is no contradiction.
*
*If it never bounces to the right, it lands in the left most bin.
*If it once bounces to the right, it lands one to the right of the left most bin.
*If it twice bounces to the right, it lands two to the right of the left most bin.
And so on.
In your example there are a total of fo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3049556",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
The rook and the bishop are moving independently on the chessboard starting at the same corner The rook and the bishop are moving independently on the chessboard starting at the same corner. What is the average number of steps until they meet again in the same corner, if we know that the bishop moves only on one quarte... | The PBS infinite series episode “Can a Chess Piece Explain Markov Chains?” can give you an insight into solving this problem.
I realize I’m several years late to the party here but the question is new to me.
To calculate how long it will take a piece to return to its starting position you take every square that piece c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3049643",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Probability of a coin falling within a tile Consider the following question:
Question
I considered leaving a border of 3 cm on each side of the tile.
Image
Now if the center of the coin falls anywhere within this region, the coin stays inside the tile. Otherwise it moves, fully or partly, outside the tile.
So favourabl... | (My previous answer was wrong - we all make mistakes, as suggested in this new response)
You seem to be correct to be concerned
Your calculation is effectively $\dfrac{(10-2\times 3)^2}{10^2}=\dfrac{4^2}{10^2}=0.16$ and looks sensible
There are two obvious ways to get $0.36 =\dfrac{6^2}{10^2}$, either as $\dfrac{(10-2\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3049711",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
References/Proof of the conjectured identity for the Stirling permutation number $\left\{{n\atop n-k}\right\}$ While working with a combinatorics problem, I conjectured that
$$ \left\{{n \atop n-k }\right\}=\sum_{p=0}^{k-1}\bigg\langle\!\!\bigg\langle{k\atop k-1-p}\bigg\rangle\!\!\bigg\rangle \binom{n+p}{2k}, $$
where ... | "Concrete Mathematics (what else?) - Eulerian Numbers" - says:
"Second-order Eulerian numbers are important chiefly because of their connection with Stirling numbers"
Eq. (6.43) therein gives
$$
\left\{ \matrix{ x \cr x - n \cr} \right\}
= \sum\limits_{\left( {0\, \le } \right)\,k\,\left( { \le \,n} \right)} {\lef... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3049932",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Stone–Weierstrass for maps $S^m\to S^n$? In the middle of page 35 of Algebraic Topology by Tammo tom Dieck, the author remarks:
If $f:S^m\to S^n$ is a continuous map, then there exists (by the theorem of
Stone–Weierstrass, say) a $C^\infty$-map $g:S^m\to S^n$ such that $\|f(x)-g(x)\|<2$, $\forall\, x\in S^m$.
Here,... | By the usual version of Stone-Weierstrass for maps $S^m \to \mathbb R$, you can approximate $f$ by a smooth $h: S^m \to \mathbb R^{n+1}$ (i.e. approximate each coordinate of $f$). But the map $p: \mathbb R^{n+1}\setminus\{0\} \to S^n$ given by $p(x) = x/\|x\|$ is smooth, so take $g = p \circ h$ if $\|f - h\| < 1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3050078",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
The Ages of Mathematician´s sons Two mathematicians meet and talk:
"Do you have a son?" asked the first mathematician.
"Yes I actually have three sons, and none of them are twins." answered the second mathematician.
"How old are they?" asked the first mathematician.
"The product of their age is equal to the month numbe... | Let $A_1,A_2,A_3$ be the ages of the sons respectively. Observe that if the month is $1,2,3,4,5,7,9,11$ then there are no solutions using the fact that there are no twins. If the month is $6,8$ or $10$ then there is a unique solutions so just by the first information it would be possible to determine the ages. Hence th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3050195",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Question regarding trees Let $(T,<)$ be tree of height $\aleph_2$ in which every level $L_\alpha$ is countable. A proof I am reading claims that for every $t\in L_{\omega_1}$ there is a $s<t$ which extends uniqueley to level $\omega_1$, that is with $\{t'\in T \mid s\le t'\}\cap L_{\omega_1}=\{t\}$. But I don't see why... | Yes, you are right.
Some people, however, use the word tree to mean that it satisfies a normality condition, by which every branch up to a limit level has a unique limit node. Your counterexample tree is not normal in this sense.
Meanwhile, perhaps the theorem you are reading is the theorem asserting that every tall ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3050330",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
An exercise on the calculation of a function of operator The operator is given by
$$A=\begin{pmatrix}
1 & 0 & 0\\
1 & 1 & 0\\
0 & 0 & 4
\end{pmatrix}$$
I have to write down the operator $$B=\tan(\frac{\pi} {4}A)$$
I calculate $$\mathcal{R} (z) =\frac{1}{z\mathbb{1}-A}=\begin{pmatrix}
\frac{1}{z-1} & 0 & 0\\
\frac{1}{(... | It would appear that there’s an error in either the problem statement or the solution.
You can check your own answer by computing this via a consequence of the Cayley-Hamilton theorem: any analytic function of $A$ can be expressed as a quadratic polynomial in $A$, i.e., $\tan\left(\frac\pi4A\right) = aI+bA+cA^2$ for ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3050497",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Solving the matrix equation How can I solve the matrix equation of the form
$$
\mathbf{SXK} + \mathbf{X} = \mathbf{Y}
$$
Here $\mathbf{S}$ and $\mathbf{K}$ are symmetric matrices, in addition $\mathbf{K}$ is a sparse symmetric matrix. $X$ is the variable. Though $\mathbf{S}$ and $\mathbf{K}$ are symmetric, it is not i... | This equation is similar to the discrete Lyapunov equation and can be solved in a similar way. Using the equality
$$ \operatorname{vec}(ABC)=(C^{T} \otimes A)\operatorname{vec}(B) $$
one obtains the system of linear equations
$$
\left( K^T \otimes S+I_{n^2} \right)\operatorname{vec}(X)=\operatorname{vec}(Y).
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3050630",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
What is the difference between a statement and sentence in mathematical logic? I have seen many (GENERAL, BEGINNER TYPE) definitions, however, the actual meaning of a sentence I have yet to find, that is non-specific to a particular domain. This would be useful since a statement is defined in terms of a sentence and is... | I would say most texts don't make a difference between the two. In fact: statement, sentence, claim, and proposition are typically all seen as the same thing: something that has a truth-value.
If a text does make a distinction, I suspect it might be between the syntactical expression that we use in order to express a c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3050772",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Quantifying how crowded points are in $d$ dimensional cube Let $x_1, \cdots, x_n$ be distinct points in the $d$ dimensional cube $[-1,1]^d$. What is a lower bound on the quantity:
$$ \sum_{1 \le j < k \le n} \frac{1}{||x_k-x_j||}$$
where $|| \cdot ||$ is the Euclidean norm.
An asymptotic answer for large $n$ is also fi... | This is an answer only up to $n = \exp(ad)$ for some $a \in \theta(1)$, but may be enough to get started: There are as many as $n = \exp(ad)$ points $y_1,\ldots, y_n$ in $\{-1,1\}^d$ that satisfy the following:
$||y_i-y_j||_1 \in \theta(d)$ for each $i \not = j$, where $|| \cdot ||_1$ denotes the Manhattan metric.
[Go... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3050869",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
How to show that $\sum_{n=1}^{\infty}\frac{\phi^{2n}}{n^2{2n \choose n}}=\frac{9}{50}\pi^2$ Given:$$\sum_{n=1}^{\infty}\frac{\phi^{2n}}{n^2{2n \choose n}}=\frac{9}{50}\pi^2$$
Where $\phi=\frac{\sqrt{5}+1}{2}$
How can I we show that the above sum is correct?
I have checked numerically, it seem correct, but i don't how t... | Replace $\phi$ with $x$. That gives a function of $x$ with its Taylor series.
Now to find a differential equation that the Taylor series obeys.
Hint:
On one hand, $d^2f/dx^2=\sum (2n+2)(2n+1)a_{n+1}x^{2n}$.
On the other hand $xdf/dx=\sum 2n a_n x^{2n}$.
Good luck with that. As a second hint, other people have sup... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3051109",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Finding prenex normal form of a formula
Find prenex normal form of the formula $(\exists x)S(x,y)\rightarrow (R(x)\rightarrow \neg(\exists u)S(x,u))$
My attempt:
*
*$(\exists x)S(x,y)\rightarrow (R(x)\rightarrow \neg(\exists u)S(x,u))$
*$(\exists x)S(x,y)\rightarrow (R(x)\rightarrow (\forall u)\neg S(x,u))$
*$(\... | Yes, that is correct, though I would break that step into two: first replace the variable, and then bring out the quantifier. So:
$(\forall u) ((\exists x) S(x,y) \rightarrow (R(x) \rightarrow \neg S(x,u))) \overset{\text{Replace variables}}\Leftrightarrow$
$(\forall u) ((\exists w) S(w,y) \rightarrow (R(x) \rightarrow... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3051267",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Does the value of the $\lim_{x \to 0-} x^x = 1$? I have the following attempt.
Let $x=-y$ then ${y \to 0+}$ as ${x \to 0-}$.
So, $\displaystyle\lim_{x \to 0-} {x}^{x}$= $\displaystyle\lim_{y \to 0+} {(-y)}^{(-y)} = \displaystyle\lim_{y \to 0+} \dfrac{1}{{(-y)}^{y}}= \displaystyle\lim_{y \to 0+} \dfrac{1}{{(-1)}^{y}.{y... | For complex values of $z$ and $w$, we have by definition
$$\begin{align}
z^w&=e^{w\log(z)}\\\\
&=e^{w\text{Log}(|z|)+iw\arg(z)}\tag1
\end{align}$$
where $\text{Log}$ is the logarithm function of real variables and $\arg(z)$ is the multi-valued argument of $z$.
Using $(1)$ reveals for $x\in \mathbb{R}$ and $x<0$
$$\begi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3051361",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Evaluate $\sum_{n=1}^{\infty} \frac{2n+1}{(n^{2} +n)^{2}}$
Evaluate $$\sum_{n=1}^{\infty} \frac{2n+1}{(n^{2}+n)^{2}}.$$
I am getting two different results by using two different methods -
First Method
The above sum can be written as
\begin{align}\sum_{n=1}^{N} (1/n^{2} - 1/(n+1)^{2})&= 1 - 1/4 + 1/4 - 1/9 \dots -1... | In context, hopefully not too trivial.
Let $f(n)=\dfrac{2n+1}{(n^2+n)^2}$, $f(n)$ is strictly decreasing.
1)Your sum $\sum_{1}^{\infty}f(n)$ is an upper sum
for the integral $\int_{1}^{\infty}f(x)dx$.
$U :=\sum_{1}^{\infty}f(n)=1$;
2)Now consider the lower sum:
$\sum_{2}^{\infty}f(n)$ for the integral.
$L := \sum_{2}^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3051479",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Prove that if $a > b > 0, p > 0$, then $a^p > b^p$
Prove that if $a > b > 0, p > 0$, then $a^p > b^p.$
As I was reading baby Rudin, this fact was a step that Rudin skipped (Theorem 3.20a), but it is not obvious to me how to prove this.
Thanks in advance.
EDIT (relevant definitions and results from exercise 6, chapter... | You are using Rudin's "Principals of Mathematical Induction" and you are doing Chapter 1, Excercise 6. Which relies very heavily on the Theorem 1.21 and the proof thereof that;
For any $b > 1$ and $n \in \mathbb N$ there is a unique positive $c$ so that $c^n =b$. We call such a $c:= b^{\frac 1n}$.
The proof makes us... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3051598",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 2
} |
Span of a Vector Space in $\mathbb{R}^3$ Consider the subspaces $W_1$ and $W_2$ of $\mathbb{R}^3$ given by
$W_1= \{(x,y,z) \in \mathbb{R}^3:x+y+z=0 \}$ and $W_2=\{(x,y,z) \in \mathbb{R}^3:x-y+z=0 \}$.
If $W$ is a subspace of $\mathbb{R}^3$ such that
*
*$W \cap W_2= \mathrm{span}\bigl\{(0,1,1)\bigr\}$
*$W \cap W_1$... | You reasoned correctly and discarded $2$ and $4$. It must be $1$, since $(1,0,-1)$ isn't orthogonal to $(0,1,1)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3051710",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Showing an Integrable function is everywhere discontinuous
Question
Let $f(x) = x^{-1/2}$ for $0<x<1$ and $0$ otherwise, $r_n$ be an
enumeration of rationals. Define
\begin{equation}
g(x) = \sum_n g_n(x) \quad \text{where} \quad g_n(x) = 2^{-n}f(x-r_n)
\end{equation}
Show that $g$ is discontinuous everywhere... | Apparently the codomain of $g$ includes $\infty$. That can in itself be okay; there are good topologies on the extended real line.
Your argument seems to lead to the the fact that there is a dense set of points $x$ where $g(x)=\infty$. However, that in itself does not make $g$ discontinuous everywhere -- as far as you ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3051816",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Tracing the path of the point A wheel of radius $R$ is rolling inside a fixed
circular cylinder of radius $2R$ as shown.
What is the trajectory followed by a point
on the rim of the wheel?
By observation, the only two points that seem to move in a straight line, are one at the centre of the cylinder and one at the com... | This orbit is called hypotrochoid and has the parametric equations
$$
x(\theta) = (R-r)\cos\theta+r\cos\left(\frac{(R-r)\theta}{r}\right)\\
y(\theta) = (R-r)\sin\theta-r\sin\left(\frac{(R-r)\theta}{r}\right)\\
$$
in the present case we have $R=2R_0$ and $r = R_0$ so we have
$$
x(\theta) = R_0(\cos\theta+\cos\theta)\\
y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3051979",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Is the natural order relation on an idempotent semiring total/linear? We know that on an idempotent semiring $R$, the natural order relation is defined as: for all $x, y\in R$, $x\leq y$ when $x+y=y$, which is clearly a partial order relation. I am unable to point out whether this relation is a total order relation too... | A distributive lattice is an idempotent semiring (with addition $\vee$ and multiplication $\wedge$), but most lattices are not totally ordered.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3052091",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Closed subsets of compact sets are compact (original proof)
Baby Rudin Theorem 2.35: Suppose $F \subset K \subset X$ where $X$ a metric space. Suppose $F$ closed relative to $K$ and $K$ compact. Then $F$ is compact.
This is my original attempt to prove this theorem.
Here, Rudin's definitions are :
*
*$p$ is a li... | Refined proof:
Suppose $F$ not compact. Then there is a cover of $F$ which has no finite subcover say $S$. Define $M=\{X\}\setminus F$. Then $M \cup S$ is a cover of $K$. But then clearly, this cover has no finite subcover hence $K$ is not compact. $\square$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3052228",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Solve linear equation I am working on some practice linear algebra problems but I am not understanding how the answers where gotten for the following problem.
Problem:
find all values of $a$ for which the resulting linear system has (a) no solution, (b) a unique solution, and (c) infinitely many solutions.
\begin{cases... | our equations: (a^2 means a * a)
x+y−z=2
x+2y+z=3
x+y+(a^2−5)z=a
general equation form:
a1*x+b1*y+c1*z = d1
a2*x+b2*y+c2*z = d2
a3*x+b3*y+c3*z = d3
matrix A:
a1 b1 c1
a2 b2 c2
a3 b3 c3
matrix B:
d1 b1 c1
d2 b2 c2
d2 b3 c3
matrix C:
a1 d1 c1
a2 d2 c2
a3 d3 c3
matrix D:
a1 b1 d1
a2 b2 d2
a3 b3 d3
x = det(B) / det(A);
y ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3052338",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
} |
Letters in Mailboxes so that none are in the right one In how many ways can 5 letters be put in 5 mailboxes such that none are placed in the right one ? I creatively thought labeling the letters & mailboxes 1-5 , it would be saying letter 1 can be placed in any of the other four . So a total of 4^5 ways .this was not ... | These are known as derangements.
Look here:
https://en.wikipedia.org/wiki/Derangement
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3052457",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Need help with integration by substitution for $\int_x^{x+1}\left(\sin\ t^2\right)dt$ I am doing exer 13 of baby Rudin Ch 6. I learned integration by substitution and by parts only recently, hence having trouble doing it.
Exercise: Define $$f(x)=\int_x^{x+1}\left(\sin\ t^2\right)dt\ .$$ Prove that $|f(x)|<\frac{1}{x}... | Since $\vert\sin t^2\vert\le 1$ you always have that $\vert f(x)\vert\le\int_x^{x+1}1\,dt=1$. To get strict inequality, take any $t_0\in (x,x+1)$ such that $\vert\sin t^2_0\vert<1$. Then by continuity you can find that $\vert\sin t^2\vert<1-\varepsilon$ in a small interval $(t_0-\delta,t_0+\delta)$ contained in $(x,x+1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3052725",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Is there a fixed point theorem I could use to solve this problem? let $E = C([0,1]),\,\,$ $K : E \to E, \,\,
(Kf)(x) = \int_0^1K(x,y)f(y)dy$
also $\|K\| \leq a < 1$
I want to prove that there for $g \in E$ there exists a unique $f_g \in E$ that satisfies the following equation :
$f_g + Kf_g = g$
which is equivalent to... | You can apply the Contraction mapping, a.k.a. Banach's Fixed Point Theorem. Given $f,h\in C([0,1])$,
$$
\|Tf-Th\|\le\int_0^1|K(x,y)|\,|f(y)-h(y)|\,dy\le\|K\|\,\|f-h\|<a\,\|f-h\|,
$$
with $0<a<1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3052823",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
$\sum_{n=1}^{\infty} \frac{(1/2) + (-1)^{n}}{n}$ converges or diverges? How to check if the series $$\sum_{n=1}^{\infty} \frac{(1/2) + (-1)^{n}}{n}$$ converges or diverges?
When $n$ is odd, series is $\sum \frac{-1}{2n}$
When $n$ is even, series is $\sum \frac{3}{2n}$
This series is similar to the series
$$\sum \frac{... | The idea is correct, but not correctly expressed. Asserting that the given series converges is equivalent to the assertion that the sequence$$\left(\sum_{n=1}^N\frac{\frac12+(-1)^n}n\right)_{N\in\mathbb N}$$converges. If it does, then the sequence$$\left(\sum_{n=1}^{2N}\frac{\frac12+(-1)^n}n\right)_{N\in\mathbb N}$$con... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3052925",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Ideal in a $C^*$ algebra Suppose $A$ is a non-unital $C^*$ algebra, $a\in A$, $I$ is the ideal generated by $a$.
In the unital case, $a=1a1\in AaA$. But in the non-unital case, how to show that $a\in A$, can $a$ be expressed by elements in $AaA$?
| I'll assume we are talking (as usual) about closed bilateral ideals. Any C$^*$-algebra has an approximate unit $\{e_j\}$: that is, $0\leq e_j$, $\|e_j\|\leq 1$, and $\lim_j e_ja=\lim_jae_j=a$ for all $a\in A$. Then
$$
a=\lim_j e_jae_j\in \overline{AaA},
$$
without even needing sums.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3053015",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Probability an ace lies behind first ace Consider a deck of 52 cards. I keep drawing until the first ace appears. I wish to find the probability that the card after is an ace.
Now, the method I know leads to the correct answer is that given the first ace, there are $48$ possible non-ace cards that can be drawn after th... | The original question only asks about the probability of the first 2 aces being consecutive, with no conditions on the other 2 aces (apart from the obvious one that they must be later in the deck). However, your second method checks for the complement of there not being two consecutive aces anywhere among the 52 cards... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3053141",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How to find the triangle area inside the parabola? Please help me understand. The parabola $C$ has cartesian equation $y^2 = 12x.$
The point $P(3p^2, 6p)$ lies on $C,$ where $p\neq0.$
*
*(a) Show that the equation of the normal to the curve $C$ at the point $P$ is
$$y + px = 6p + 3p^3$$
This normal crosses the curve... | For $p=2$ we have: $P=(12,12)$, $Q=(27,-18)$, $S=(3,0)$. Moreover, line $PQ$ intersects the $x$-axis at $R=(18,0)$. It follows that triangles $PRS$ and $QRS$ have base $RS=15$ in common and altitudes $P_y=12$, $|Q_y|=18$, so that:
$$
area_{PSQ}=area_{PRS}+area_{QRS}={1\over2}15\cdot12+{1\over2}15\cdot18=225.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3053268",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How to find interval where function $f(x)=x+\frac{1}{x^{3}}$ is one to one\injective? How to find interval where function is one to one $f(x)=x+\frac{1}{x^{3}}$ ?(graphically or algebraically analytically)
Let $f(x) =f(y)$ this gives $ (x-y)(\frac{(xy)^{3}-y^{2}-x^{2}+xy}{(xy)^{3}})=0 $
I don't know how to proceed fur... | Hint :
$$f'(x) = 1 - \frac{3}{x^4} \implies \begin{cases} f'(x) > 0, \; x \in (-\infty,-3^{1/4})\cup(3^{1/4},\infty) \\ f'(x) <0, \; x \in (-3^{1/4},0)\cup(0,3^{1/4}) \end{cases}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3053339",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
Fake proof of differentiability It's a theorem that if $f\colon U\subset\Bbb R^n\to \Bbb R^m$ has the property that each of the partial derivatives $\partial_if_j$ exist and are continuous $p\in U$, then $f$ is differentiable at $p$. When I was trying to prove this, I came up with the following "proof" which doesn't us... | In short, if you don't assume that the $\partial_i f_j$ are continuous then you can't assume that $f_j$ is differentiable at $p$.
You only know that all partial derivatives of $f_j$ exist, but you need continuity to guarantee that $f_j$ is actually differentiable (that's the $m=1$ case of the theorem you talk about).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3053412",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Evaluate:$S_{n}=\binom{n}{0}-\binom{n-1}{1}+\binom{n-2}{n-3}-\binom{n-3}{n-6}+.......$ If$$S_{n}=\binom{n}{0}-\binom{n-1}{1}+\binom{n-2}{n-3}-\binom{n-3}{n-6}+.......$$
Does $S_{n}$ have a closed form.
My Attempt
$$S_{n}=\binom{n}{0}-\binom{n-1}{n-2}+\binom{n-2}{2}-\binom{n-3}{3}+.......$$
$$S_{n}=\binom{n}{n}-\binom{n... | Here is a combinatorial solution.
Consider tilings of an $n\times 1$ rectangle with squares and dominos. There are $\binom{n-k}{k}$ such tilings which use exactly $k$ dominos. Your sum counts all such tilings, where those with an even number of dominos are counted positively, and those with an odd number of dominos are... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3053514",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
$n$ players are each dealt two cards — what's the probability that $k$ of them have a pair? Each of $n\leq26$ players is dealt $2$ cards from a standard $52$-card poker deck. What is $\textrm{P}\left(n,k\right)$, the probability that exactly $k$ of the $n$ players have a pair?
(A pair is a hand like $8 \clubsuit, 8 \he... | First, let us compute $f(m)$, the number of ways to deal two cards to each of $m$ people which are all pairs. We can do this using a computer as follows. If we further let $f(m,r)$ be the probability that all $m$ people have pairs when dealt from a deck of $4r$ cards, then
$$
f(m,r) = r\binom42\big(f(m-1,r-1)+(m-1)f(m-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3053726",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Not necesssarily locally compact and locally compact difference
A topological space is said to be locally compact if each point $x\in X$ has at least one neighbourhood which is compact.
If $f$ is continuous open mapping of a locally compact space $(X,\tau)$ onto a topological space $(Y,\tau_1)$ then $(Y,\tau_1)$ is ... | 1) The second question is asking you for an example of topological spaces $X$ and $Y$ and of a function $f\colon X\longrightarrow Y$ such that:
*
*$f$ is continuous;
*$f$ is surjective;
*$X$ is locally compact;
*$Y$ is not locally compact.
2) You did not justify the assertion that $f(U)$ is a neighborhood of $y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3053996",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Value of k to give matrix infinite, 0, 1 solutions. I have a question that goes:
For which values of the constant $k$ does the system of equations below have:
*
*a unique solution,
*no solutions at all,
*infinitely many solutions?
$$
\begin{cases}
x &- 3y & &= 6\\
x & &+ 3z &= -3\\
2x &+ ky &... | You have
$$
\begin{pmatrix}
1 & -3 & 0 & 6 \\
1 & 0 & 3 & -3 \\
2 & k & 3-k & 1
\end{pmatrix}
\to
\begin{pmatrix}
1 & -3 & 0 & 6 \\
0 & 3 & 3 & -9 \\
0 & k+6 & 3-k & -11
\end{pmatrix}
\to\\
\begin{pmatrix}
1 & 0 & 3 & -3 \\
0 & 1 & 1 & -3 \\
0 & 0 & 3-k-(k+6) & -11 + 3(k+6)
\end{pmatrix}
\to \\
\b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3054104",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How do I convert $y= 2x^{2} + 16x$ into the vertex form (i.e. $y=a(x-h)^{2}+k$)? I tried looking up the "process" of solving that equation, but I couldn't really find the exact way to solve it.
Isolating the $2$ from $2x^2$ might be one of the way, but I couldn't exactly find out what I would have to do after that.
Tha... | While Dr. Sonnhard Graubner's answer is valid, I'd like to present a more intuitive approach.
Recall: the vertex form of a parabola is given by $y = a(x - h)^2 + k$, for vertex $(h,k)$. For the sake of argument, we can expand that form by foiling the squared term:
$$y = ax^2 - 2hax + ah^2 + k$$
We seek to write $y = 2x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3054264",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Mistake in Billingsleys book? This is about an exercise in Billingsley's book Probability and measure.
Exercise 2.15: On the field $\mathscr B_0$ in $(0,1]$ define $P(A)$ to be $1$ or $0$ according as there does or does not exist some positive $\epsilon_A$ (depending on $A$) such that $A$ contains the interval $(\frac{... | Your mistake: Recall that $\mathscr{B}_0$ is defined as the set of "finite disjoint unions of intervals in $(0,1]$." The issue with your counterexample is simply that the $A,B$ you use do not belong to $\mathscr{B}_0$.
Now, take any two disjoint $A,B\in\mathscr{B}_0$: by assumption, there exist $n,m\geq 1$ and disjoin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3054351",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Exterior Covering Number of $\epsilon /2$ is greater of equal than Covering Number of $\epsilon$ Definition $(\epsilon -Net)$ :let $(T,d)$ be a metric space .Consider a subset $ K \subset T$ and let $\epsilon >0$, A subset $N \subset K $ is called $\epsilon -Net$ of $K$ if every point in $K$ is within a distance $\eps... | Suppose that $$\bar B(x_1, \epsilon / 2), \ \dots, \ \bar B(x_{N^{\rm ext}}, \epsilon / 2)$$ is an external covering of $K$ of minimal size.
For each $i \in \{ 1, \dots, N^{\rm ext}\}$, there exists a $k_i \in K$ that is contained in $\bar B(x_i, \epsilon / 2)$. (Otherwise $\bar B(x_i, \epsilon / 2)$ would be redundant... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3054478",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.