Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Linear codes and check matrices. Two linear codes are defined as the following subspaces of $\Bbb B_7$:
$C_1$ with dimension $2$ and basis $\{1010101,\ 0101010\}$
$C_2=\text{ker}(H)$ where $H=\left (\!\begin {array}{ccccccc} 0&0&0&1&1&1&1\\ 0&1&1&0&0&1&1\\ 1&0&1&0&1&0&1 \end {array}\!\right )$.
Just wanted to check, to... | $C_1$ has dimension 2 and consists of the zero vector, the two basis vectors and the all-1 vector (by adding the two basis vectors).
$C_2$ has dimension $m=4 = 7-3$, since the kernel has dimension 3 (as can be seen from the check matrix) and the ambient space is $\Bbb Z_2^7$.
| {
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"url": "https://math.stackexchange.com/questions/3806012",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finding remainder of $123^{456}$ divided by 88 using Chinese Remainder Theorem I tried using Chinese remainder theorem but I kept getting 19 instead of 9.
Here are my steps
$$
\begin{split}
M &= 88 = 8 \times 11 \\
x_1 &= 123^{456}\equiv 2^{456} \equiv 2^{6} \equiv 64 \equiv 9 \pmod{11} \\
y_1 &= 9^{-1} \equiv 9^9 \equ... | $y_1$ should've been the inverse of $8\pmod{11}$, not of $9\pmod{11}$, so $y_1=7$.
Similarly, $y_2$ should've been the inverse of $11\pmod 8$, not of $1\pmod 8$, so $y_2=3$.
Therefore, the result is: $9\times\frac{88}{11}\times \color{red}{7}+1\times\frac{88}{8}\times \color{red}{3}=537\equiv 9\pmod{88}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3806122",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
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Does a curve (differentiable manifold of dim 1) always have a parametrization? I don't know much about this field, so this is a basic question. I think there are 2 similar basic concepts referring to curve:
*
*A $C^k$ differential curve, can be defined as a differentiable $C^k$ manifold of dimension 1. This is what o... | When you say
Now I think it is obvious that the trajectory of a $C^k$-parametric curve (ie $f(I)$) is a $C^k$ differential curve...
this is in fact not correct. Consider for example a lemniscate.
The converse is not true either. Consider a manifold of dimension one with two connected components.
| {
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"url": "https://math.stackexchange.com/questions/3806230",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is the product of source and target maps of a Lie Groupoid a submersion? Let $G:= (G_1 \rightrightarrows G_0)$ be a Lie Groupoid. By definition, we know that source $s$ and target $t$ are submersion. Now define $(s,t):G_1 \rightarrow G_0 \times G_0$ as $\gamma \mapsto (s(\gamma),t(\gamma))$.
My Question is the followin... | In general, the map $(s,t):G_1\rightarrow G_0\times G_0$ need not be a submersion.
Suppose $M$ is a positive dimensional manifold and consider the Lie groupoid $M \rightrightarrows M$ where we think of the objects as being $M$, and for each $m\in M$, the only arrow at $m$ is the identity arrow (so the collection of arr... | {
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"source": "stackexchange",
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Simple space-filling curve Here are six iterations of Hilbert space-filling curve.
Isn't there a simpler space-filling curve?
For example $16$ iterations of this curve:
Isn't it also a space-filling curve?
*
*If it is not then why?
*If it is then why they (mathematicians) did not use the simplest possible curve? ... | The images you provided don't lead to the construction of a space-filling curve. This is due to the fact that if you take $\{f_n\}$ to be the sequence of functions defining the iterations that you have provided, normally we'd call $f:= \lim_{n\rightarrow \infty} f_n$ our resulting space-filling curve. However, in this ... | {
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Swapping modulus and argument in polar coordinates Let $y=f(x)$ in Cartesian coordinates. Swapping the $x$ coordinate with the $y$ coordinate has the effect of reflection about the axis $y=x$.
But if $r=f(\theta )$ in polar coordinates (where $r$ is the modulus and $\theta$ is the argument), what happens when we swap $... | Polar coordinates we call well known mapping $\mathbb{R}^2 \to \mathbb{R}^2$, from $(x,y)$ to $(\theta, r)$ using formulas
$x = r\cos \theta$, $y = r\sin \theta$, $r \geqslant 0,\theta \in [0, 2\pi) $.
As to plane $(\theta, r)$, then it is usual cartesian coordinates, usual $\mathbb{R}^2$, and you can think about it ex... | {
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"source": "stackexchange",
"question_score": "2",
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What is relationship between beta and binomial distributions in Bayesian inference I came across this question:
Suppose we are giving two students a multiple-choice exam with 40 questions, where each question has four choices. We don't know how much the students have studied for this exam, but we think that they will d... | The point is that if the prior is a beta distribution and the likelihood comes from a binomial distribution, then the posterior is a again a beta distribution.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3806844",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Integrate $ \int \frac{1}{\sin^{4}x+\cos^{4}x}dx $ Show that$$ \int \frac{1}{\sin^{4}(x)+\cos^{4}(x)}dx \ = \frac{1}{\sqrt{2}}\arctan\left(\frac{\tan2x}{\sqrt{2}}\right)+C$$
I have tried using Weierstrass substitution but I can't seem to get to the answer... Should I be using the said method or is there another way I c... | \begin{align}
\int \frac{dx}{\sin^{4} x+\cos^{4} x}
&= \int \frac{dx}{\frac14(1-\cos2x)^2+\frac14(1+\cos2x)^2}\\
&= \int \frac{2dx}{1+\cos^22x}=\int\frac{2\sec^22xdx}{2+\tan^22x}\\
&=\frac1{\sqrt2}\int\frac{d(\frac{\tan2x}{\sqrt2})}{1+(\frac{\tan2x}{\sqrt2})^2}=\frac{1}{\sqrt{2}}\arctan\frac{\tan2x}{\sqrt{2}}+C
\end{a... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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$M_n(k)$ is an indecomposable algebra Let $A$ be a $k$-algebra. The following are equivalent:
(i) $A$ is indecomposable as a $k$-algebra.
(ii) $A$ is indecomposable as an $A-A$-bimodule.
(iii) The idempotent $1_A$ is primitive in $Z(A)$.
Then the author says later that $M_n(k)$ is an indecomposable algebra. Why? Apprar... | The identity matrix isn’t primitive in $M_n(k)$, but it is primitive in $Z(M_n(k))=k$, which is what the third thing says.
Any simple $k$-algebra $A$ has to be indecomposable, of course, because a nontrivial central idempotent $e$ would create a nontrivial ideal $eA$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3807100",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Elements of quotient group with $\mathbb Z$-basis Abelian free group $G, H$ have rank $2$ and $G$ has $\Bbb Z$-basis $x, y$, if $H$ has $\Bbb Z$-basis
$$2x+y, 2x-3y$$ then what are the elements of $G/ H$ ?
I am new to the topics, so don't know how to start. The question is motivated from the following excerpt of the bo... | I assume that $G$ is a free Abelian group over a two-element set $\{x,y\}$ and $H$ is the subgroup of $G$ generated by $2x+y$ and $2x-3y$. It is easy to check that $H$ has rank $2$. The theorem on subgroups of a finitely generated free Abelian group (see, for instance, [§20, Kur]) implies that there exist bases $\{u_1,... | {
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Two inequalities for proving that there are no odd perfect numbers? Let $n$ be a natural number. Let $U_n = \{d \in \mathbb{N}| d|n \text{ and } \gcd(d,n/d)=1 \}$ be the set of unitary divisors, $D_n$ be the set of divisors and $S_n=\{d \in \mathbb{N}|d^2 | n\}$ be the set of square divisors of $n$.
The set $U_n$ is a ... | *
*$p^{2k+1} > \dfrac{p^{2(k+1)} - 1}{2(p-1)}$ equals to $(p-2)p^{2k+1} + 1 > 0$ for $p \ge 2$ and $k \ge 0$
*$p^{2k} > \dfrac{p^{2k+1} + p^{k+1} - p ^ k - 1}{2(p-1)}$ equals to $(p^k-1)((p-2)p^k-1) > 0$ for $p > 2$ and $k \ge 1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3807399",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Finding the limit: $\lim_{x\to \infty}\frac{1}{2}x\sin {\frac{180(x-2)}{x}}$ While investigating a problem, I came across a function: $$f(x) = \frac{1}{2}x\sin {\frac{180(x-2)}{x}}$$ When looking at the function in Desmos (I was checking my proof), I discovered that $$\lim_{x\to \infty}\frac{1}{2}x\sin {\frac{180(x-2)... | You really shouldn't work in degrees; more specifically, the sin function itself is defined (despite what you might have learned in high school) with 'radian' arguments. Formulae like $e^{ix}=\cos x+i\sin x$, or $\frac{d}{dx}\sin x=\cos x$, rely on it. There's another formula that relies on it that's the critical one h... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3807488",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Eigenvectors of action span the representation Let $V$ be a complex finite dimensional vector space and $\rho:S_3\to \text{GL}(V)$ be a representation of the symmetric group $S_3$. Let $A_3$ be the alternating subgroup of $S_3$; let $\tau$ be a generator of $A_3$ (since $A_3$ is cyclic.). I want to prove that $V$ is s... | Recursive proof on $dim(V)$, since $S^3$ is finite, there exists an Hermitian product of $V$ invariant by the action of $V$. Let $c$ be an eigenvalue of $\tau$, and $V_c$ the associate eigenspace, $V'$ the orthogonal of $V_c$ is invariant by $\tau$. If $V_c\neq V$, apply the recursive hypothesis to $V'$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3807616",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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recurrence relation where $c_n = c_{n-1} + 2c_{n-2}$
A sequence $(c_n)$ is defined recursively as follows: $c_0 = 1, c_1 = 1, $ and $c_n = c_{n-1} + 2c_{n-2}$ for $n\geq 2$. We use $[x^n]g(x)$ to denote the coefficient of $x^n$ of the polynomial $g(x).$ Show that $c_{2n} = [x^{2n}] \dfrac{1}2\left(\dfrac{1}{1-x-2x^2}+... | Using generating functions, I obtained the expression for $G(z) = \sum_n c_n z^n$:
$$
G(z) = \frac{c_0(1-z)+c_1z}{(1+z)(1-2z)}
$$
Now you need to expand the expression with partial fractions, you will obtain on RHS two expressions of the form
$$
G(z) = \lambda_1\sum_{z=0}^{\infty}(-1)^kz^k + \lambda_1\sum_{z=0}^{\infty... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Is this discrete analogue of Fatou's lemma valid I was curious if this discrete analog of Fatou's lemma is valid:
\begin{align}
\sum_{j=1}^\infty \liminf_{k \rightarrow\infty} a_j(k) \leq \liminf_{k \rightarrow\infty} \sum_{j=1}^\infty a_j(k) ,
\end{align}
where $a_j(k)$ is a doubly indexed sequence of real numbers.
... | An infinite sum can be realized as an integral w.r.t counting measure on the integers. So the inequality holds in the non-negative case. Fatou's Lemma requires non-negativity and and the inequality is false without it.
$a_j(k)=-1$ for $j=k$ and $0$ for $j \neq k$ gives a counter-example.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3807824",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Let $ABC$ be triangle with sides that are not equal. Find point $X$ on $BC$ From following conditions. Let $ABC$ be triangle with sides that are not equal. Find point $X$ on $BC$ such that $$\frac{\text{area}\ \triangle{ABX}}{\text{area}\ \triangle{ACX}}=\frac{\text{perimeter}{\ \triangle{ABX}}}{\text{perimeter}{\ \tri... | Obviously, the stated condition means that
$\triangle ABX$ and $\triangle AXC$
must have the same radius of the inscribed circle.
That means $AX$ must be the incircle bisector of $\triangle ABC$,
see
Incircle bisectors and related measures.
For $A$, $B$, $C$
ordered in positive (counterclockwise) direction,
$|BC|... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3807936",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Integrating floor functions without known limits Suppose we have$$\frac{\displaystyle\int_0^n{\lfloor x \rfloor}\,dx}{\displaystyle\int_0^n{\lbrace x \rbrace }\,dx}$$ $n \in I$
where $\lfloor \cdot\rfloor$ and $\lbrace \cdot\rbrace $ represent the floor function and fractional part function.
The usual method of splitti... | Note that
$$\int_{0}^{n}\lfloor x\rfloor\,dx=\sum_{k=0}^{n-1}k\bigg[\int_{k}^{k+1}\,dx\bigg]=\frac{n(n-1)}{2}$$
Also note that $\{x\}=x-\lfloor x\rfloor$ if $x\geq 0$. The case for $x<0$ is $\{x\}=x-\lceil x\rceil$ and is easily modified from this. So, $$\int_{0}^{n} \{x\}\,dx=\int_{0}^{n}x-\lfloor x\rfloor\,dx=\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3808039",
"timestamp": "2023-03-29T00:00:00",
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Spivak's Calculus Chapter $7$ $15$b) Prove $f$ is bounded below Q15) Suppose that $\phi$ is continuous and $\lim_{x \rightarrow \infty}(\frac{\phi(x)}{x^n})=0=\lim_{x \rightarrow -\infty}(\frac{\phi(x)}{x^n})$
b) Prove that if $n$ is even, then there is a number $y$ such that $y^n + \phi(y) \leq x^n + \phi(x)$ for all ... | Hint : The function $$x \mapsto x^n + \phi(x)$$
is continuous and tends to $+\infty$ when $x$ tends to $\pm \infty$. It is sufficient to ensure that it has a minimum.
| {
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"url": "https://math.stackexchange.com/questions/3808143",
"timestamp": "2023-03-29T00:00:00",
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Solving for $p$ binomcdf$(n,p,k)$=$x$ In Binomial Probability with Unknown p, this is done for a case in which $n=0$, which ends up being a simple solution. With $n > 0$, things aren't as simple, trying to naively directly solve gets you higher and higher power equations.
E.g. solving $\text{binomcdf}(17,p,8) = 0.3$ (... | As Felix Marin points out, we have bounds on when we have $\operatorname{binomcdf}(n,p,k)-t$ changes sign given by $p=0$ and $p=1$, so bracketing methods can safely be tried to get fast and guaranteed convergence. You may want to try Brent's method, Chandrupatla's method, or even the Newton-Raphson method since the der... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3808215",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is the convex combination realizing the convex envelope unique? Let $F:[0,\infty) \to [0,\infty)$ be a continuous function satisfying $F(1)=0$, which is strictly increasing on $[1,\infty)$, and strictly decreasing on $[0,1]$. Suppose also that $F|_{(1-\epsilon,1+\epsilon)}$ is convex for some $\epsilon>0$. Suppose tha... | Take $c = 1/2$,
$$x_1=1/2 -a,\quad y_1=1/2 +a,\\ x_2=1/2 -b, \quad y_2=1/2 +b,$$
where $a<b<1/2$. You can find a function $F$ satisfying all the hypotheses such that $\hat F(c)=1-c< F(c)$, and
$$ \hat F(x) = F(x)=1-x,\quad x = x_1,x_2,y_1,y_2$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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prove the angle of sum of vectors is always in the middle For two planar vectors $v_1, v_2$, and they are in the same quadrant.
Define $\angle(v_1)$ as the angle between $v_1$ and positive x-axis. And $\angle{v_1} \lt \angle(v_2)$.
Their sum $v_3 = v_1+v_2$, would this be true $\angle(v_1) \lt \angle(v_3) \lt \angle(v_... | Hint (elaborating on Tassle's comment):
You have that
$$\tan\angle v_1=\frac{a_1}{b_1},\ \tan\angle v_2=\frac{a_2}{b_2},\ \tan\angle v_3=\frac{a_1+a_2}{b_1+b_2}.$$
Since the $\tan$ function is monotonically increasing, it suffices to show that
$$\frac{a_1}{b_1}<\frac{a_1+a_2}{b_1+b_2}<\frac{a_2}{b_2}.$$
Since we know t... | {
"language": "en",
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Why do engineers use derivatives in discontinuous functions? Is it correct? I am a Software Engineering student and this year I learned about how CPUs work, it turns out that electronic engineers and I also see it a lot in my field, we do use derivatives with discontinuous functions. For instance in order to calculate ... | If you consider that k and n are continuous variables, you obtain a continuous function that may be derivated and is entangled to the initial discontinuous function. The real problem is if the extreme point of this function can approximate the extreme point of the discontinuous one. The error may be evaluated with the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3808684",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "72",
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How many $3$-digit numbers have a digit sum of $21$? I have the following question:
How many $3$-digit numbers have a digit sum of $21$?
I can count it by taking separate cases of three digit numbers such that their digit sum is $21$ & by calculating all possible numbers that can be made from each case and by adding ... | $$a+b+c=21\implies(9-a)+(9-b)+(9-c)=6$$
Apply stars and bars to $x+y+z=6$ to get $\binom{8}{2}=28$ solutions.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3808952",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Differentiate two times a simple expression Let's say I have:
$$ y = -\frac{1}{\tan(x)} $$
How do you get a relation between $\mathrm{d}^2x$ and $\mathrm{d}^2y$ ?
What I know:
$$\mathrm{d}y = \frac{\mathrm{d}x}{\sin(x)^2} $$
| Hint: $$y^{(n)}(x)=\frac{d^ny(x)}{dx^n} \iff d^ny(x)=y^{(n)}(x)\space dx^n.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3809082",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finding a closed form expression of a sequence that is defined recursively via a definite integral Consider the following series function that is defined recursively by the following definite integral
$$
f_n(x) = \int_0^x u^n f_{n-1}(u) \, \mathrm{d}u \qquad\qquad (n \ge 1) \, ,
$$
with $f_0 (x) = \operatorname{erf}(x... | Apparently you made some index mistakes...
If $$f_n(x)=u_n x^{k_n}+O\left(x^{k_n+1}\right)$$ then it can be easily seen
$$k_{n+1}=k_n+n+2$$
$$u_{n+1}=\frac{u_n}{k_n+n+2}$$
Assuming that you are using the unnormalised error function $\text{erf}(x)=x+O(x^3)$, we have the initial conditions $k_0=1$, $u_0=1$.
Solving the f... | {
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If $P(x)=\sum_{i=0}^da_i\left(\prod_{j=i}^{d+i-1}(x+j)\right)$ is linear, what is its constant term? Question: Fix $d,m\in\mathbb{N}$ with $0\leq m\leq d$ and define
$$P(x)=\sum_{i=0}^da_i\left(\prod_{j=i}^{d+i-1}(x+j)\right),$$
where each $a_i$ is a constant, $a_m=0$. Suppose that, after expansion, $P(x)=c-x$ for some... | Some thoughts
Clearly, $d\ge 1$.
Let $x = -(m-1), -m, -(m+1), \cdots, -(d-1)$ respectively to get
\begin{align}
P(1-m) &= c + m - 1, \\
P(-m) &= c + m, \\
P(-m - 1) &= c + m+1, \\
&\cdots\cdots\\
P(-d+1) &= c + d - 1.
\end{align}
Then we have (weighted sum of the equations above)
$$\sum_{k=0}^{d-m} P(-m - k + 1)(-1)^k\... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Locally countable implies countably tight?
*
*Locally countable: The space has a base of open sets, each with countable cardinality.
*Countably tight: for each A⊆X and each $p\in\overline{A}$ there is a countable subset $D\subseteq A$ such that $p\in \overline D$.
Does locally countable imply countably tight?
| Yes, if $p \in \overline{A}$ and $U$ is open containing $p$, then $p \in \overline{U \cap A}$. So if $U$ is chosen to be countable, $U \cap A$ is as required for showing countable tightness.
| {
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"answer_id": 1
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How to prove the union bound with Markov's inequality? We have events $B_1$, $B_2$, $\dots$, $B_t$. Prove $\Pr\left(\bigcup_{i=1}^t B_i\right) \le \sum_{i=1}^{t}\Pr(B_i)$.
Wikipedia proves by induction and I also understand this inequality intuitively, that is when summing all the events you're computing the overlapped... | Hint: take $X = 1_{B_1} + \dots + 1_{B_t}$. Note that $\bigcup_{i=1}^t B_i = \{X \ge 1\}$, so use Markov's inequality to estimate the probability of the latter event.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3809886",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Can a Normal Distribution be specified by its mean and cubic deviation? I recently read that:
a normal distribution is completely specified by its mean and standard deviation.
That makes a lot of sense. But I was wondering isn't it also true that it could be completely specified by its mean and the cubic deviation? O... | From the Wikipedia entry, we know that the $p$-th central absolute moment of a $N(\mu,\sigma^2)$ random variable $X$ is
$$E[|X-\mu|^p]=\sigma^p\frac{2^{p/2}\Gamma(\frac{p+1}2)}{\sqrt\pi}.$$
If we know this number, and know $p$, we can determine $\tau=\sigma^p$
and then determine $\sigma=\tau^{1/p}$.
It might seem parad... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3809982",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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random walk inside a square (probability of escape before returning) My question comes from Exercise 9.7. of the book "Markov Chains and Mixing Times (2nd edition)" written by David A.Levin and Yuval Peres. Specifically, let $B_n$ be the subset of $\mathbb{Z}^2$ in the box of side length $2n$ centered at $0$. Let $\par... | I think I have something, if I dindn't make any mistake.
$\mathbb{P}_0\{\tau_{\partial B_n} < \tau^+_0\} = \sum_{k \in \mathbb{N} } \mathbb{P}\{\tau_{\partial B_n} < \tau^+_0 |\tau^+_0 =k \} \mathbb{P}\{ \tau^+_0 =k \} = \sum_{k \geq n } \mathbb{P}\{\tau_{\partial B_n} < \tau^+_0 |\tau^+_0 =k \} \mathbb{P}\{ \tau^+_0 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3810184",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What category is the universal property of the Free Group a diagram in? wikipedia says that the free group is defined by a universal property:
The free group $F_S$ is the universal group generated by the set $S$. This can be formalized by the following universal property: given any function $f$ from $S$ to a group $G$... | As the definition mentions, $f$ and the unnamed inclusions are just functions while $\varphi$ is a group homomorphism. Hence the diagram is not in $\mathbf{Grp}$, nor actually in $\mathbf{Set}$ (in the sense that the diagram in $\mathbf{Set}$ would not force $\varphi$ to be a group homomorphism).
The construction gives... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3810300",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Minimum distance among a set of points in COVID-19 times Students going back to schools made me think of the following (non-trivial, I think) problem.
Problem: How to arrange students in a classroom so that they keep a distance of $1.5$ meters from each other in such a way that there is room for as many students as po... | The usual formulation of this problem is that you are given a fixed shape of classroom and are supposed to maximize $n$. Imagine you extended the classroom by $0.75$ metres in each direction, then you would be able to place a circle of radius $0.75$ on each student without the circles overlapping. Vice versa, any nonov... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Expected number of days before all magical seeds become apple trees This is a question that I came across recently.
At the end of day $0$, $6$ magical seeds are planted. On each day following it, each seed has a chance to magically transform into an apple tree with a probability of $\frac{1}{2}$. The outcomes of seeds ... | Let $X_i$ be the r.v. corresponding to the number of the day where the $i$-th seed become an apple tree. Obviously, for all $i=1, ..., 6$ and for all $n \geq 1$,
$$\mathbb{P}(X_i = n) = \frac{1}{2^n}$$
What you are asked to compute is
$$\mathbb{E}(\max(X_i))$$
But for all integer $n \geq 1$,
$$\mathbb{P}(\max(X_i)=n) =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3810512",
"timestamp": "2023-03-29T00:00:00",
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How to maximize $f(x,y) = 3\sin x + 4\cos y$? Hello for an linear algebra course I have the question as seen below
*Among all the unit vectors
$$
u=\left(\begin{array}{l}
u_{1} \\
u_{2} \\
u_{3} \\
u_{4}
\end{array}\right) \in \mathbb{R}^{4}
$$
find the one for which the sum
$$
3 u_{1}-2 u_{2}+4 u_{3}-u_{4}
$$
is max... | $u$ is a unit vector means $||u|| = \sqrt{u_1^2 +u_2^2 + u_3^2 + u_4^2} = 1$. Consider it as a constraint on the vector you seek $u$ in the form
$$
g(u_1,u_2,u_3,u_4) = 0\qquad {\rm i.e. } \qquad \sqrt{u_1^2 +u_2^2 + u_3^2 + u_4^2} - 1 =0
$$
the function you have $f(u_1,u_2,u_3,u_4) = 3u_1 - 2u_2 + 4u_3-u_4$ must be ma... | {
"language": "en",
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For what values of $k$ is there a perfect $x^p$ in {$n, n+1, n+2, ... ,kn$}? I was thinking about this the other day, and I can't seem to find the answer.
It is a fairly simple proof by induction to show that for all $n ∈ ℕ$, there is a perfect square in ${n, n+1, n+2, ... , 2n}$.
I am trying to generalize this questio... | For $n = 1$, since $1$ is a $p$'th power, any $k$ will do. For $n = 2$, the next largest $p$'th power if $2^p$, so
$$kn \ge 2^p \implies k \ge 2^{p-1} \tag{1}\label{eq1A}$$
I will show the minimum allowed value of $k = 2^{p-1}$ always works. For any $2 \le n \le 2^p$, this value of $k$ works. Also, if $n = m^p$ for any... | {
"language": "en",
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Steps for solution of $\frac{dy}{dx}-y^2=\frac{a}{x}+b$ According to Wolfram,
$$\frac{dy}{dx}-y^2=\frac{b}{x}+a$$
The equation above has the solution;
Where $a$ and $b$ are constants. $_1F_1(a;b;x)$ and $U(a,b,x)$ are the Kummer confluent hypergeometric function and confluent hypergeometric function of the second kind... | What is making problem for $$\frac{dy}{dx}-y^2=\frac{b}{x}+a$$ is $a$. For $a=0$, you would get the "simple"
$$y=\frac{c_1 J_1\left(2 \sqrt{b} \sqrt{x}\right)+\sqrt{b} \sqrt{x} \left(\left(c_1-2\right)
J_0\left(2 \sqrt{b} \sqrt{x}\right)-c_1 J_2\left(2 \sqrt{b}
\sqrt{x}\right)\right)}{2 \left(1-c_1\right) x J_1\l... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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prove or disprove: if $y'=y^2-\cos(x)$ then any solution diverges in a finite time given the following ode, prove or disprove:
*
*if $y'=y^2-\cos(x)$ then any solution diverges in a finite time.
"diverge" means, that there is a point $x$ where the solution isn't continuous (there and thereafter).
if $y(0)>1$ or $y(... | Gosh -- this is hard! I've tried plotting a streamline plot of the ODE, which seems to confirm finite-time blowup. The horizontal axis is the $x$-coordinate; the vertical axis is the $y$-coordinate.
As a rough first argument, you can say that $y^2 + \cos(x) \approxeq y^2$ for $y$ large. More precisely, as $y \to \inft... | {
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what is the purpose of number system conversions e.g decimal to base 5? I'm learning Number system conversion youtube. so far I know there are decimal, binary, octal, and Hex numbers. There is a purpose behind converting decimal to binary But what is the purpose of converting decimal to base 5 number?
| Not much. The only argument for base $10$ is that we are used to it, which is very strong. Binary is useful for computer processing, but leads to very long expressions for numbers. Years ago some computers would compress binary into octal, which cuts the number of digits by a factor $3$. I worked on a CDC6400 that ... | {
"language": "en",
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Möbius transformations of finite order I saw this question, which brought up the question: can we classify all the Möbius transformations (with complex coefficients) of finite order? In particular, do these only consist of the rotations? I sense symbolic manipulation isn't the way to go about this—and indeed didn't get... | Every Möbius transformation which is not the identity is conjugate to either a translation
$$
z \to z + a \, \quad (a \in \Bbb C, a \ne 0)
$$
or a complex-linear map (rotation/dilation)
$$
z \to \lambda z \, \quad (\lambda \in \Bbb C, \lambda \ne 0, 1)
$$
depending on whether it has one or two fixed points. Translati... | {
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How to prove that the elasticity of the revenue function is $E_R(p)=\frac{E_R(q)}{E_R(q)-1}$? The Problem
Let the demand function be $ap+bq=k$.
Prove that this equation (elasticity of revenue) is true:
$$E_R(p)=\frac{E_R(q)}{E_R(q)-1}$$
Definitions
Demand Function
The Demand Function is defined as the relation between... | Note you may equivalently define elasticity (using your notation) as
$$E_f(x)=\frac{d \log f(x)}{d \log x}.$$
Thus,
$$\small E_R(q)=\frac{d \log pq}{d \log q}=\frac{d \log pq}{d \log p}\frac{d \log p}{d \log q}=\frac{d \log pq}{d \log p}\left(\frac{d (\log p+\log q)}{d \log q}-1\right)=\frac{d \log pq}{d \log p}\left(\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3811532",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If a left module M is co-isosimple and semi-Hopfian, then M is simple. I have tried to prove following theorem;
The followings are equivalent for a left module M:
*
*M is simple,
*M is co-isosimple and co-regular,
*M is co-isosimple and semi-Hopfian,
*M is co-isosimple and discrete.
So far, I have proved (1$\Rig... | Let $M$ be co-isosimple and semi-hopfian, and $N$ be any submodule not equal to $M$.
Firstly of course, $M\cong M/N$. But that means the composition of the projection $\pi:M\to M/N$ with that isomorphism is an onto endomorphism of $M$. Therefore its kernel (which is $N$) is a summand of $M$. This shows that $M$ is a se... | {
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Limit of multivariable function $f(x,y) = {(x^2+y^2)}^{x^2y^2}$ $$f(x,y) = {(x^2+y^2)}^{x^2y^2}$$
I need to find the limit at (0,0) point
I applied the exponent rule and got $$e^{x^2y^2ln(x^2+y^2)}$$
and now with chain rule, I need to find the limit of $${x^2y^2ln(x^2+y^2)}$$
and how? :D
There isn't L'Hôpital's rule fo... | We have that
$$ {(x^2+y^2)}^{x^2y^2}=e^{x^2y^2 \log(x^2+y^2)} \to 1$$
indeed
$$x^2y^2 \log(x^2+y^2)=(x^2+y^2) \log(x^2+y^2) \frac{x^2y^2}{x^2+y^2} \to 0\cdot 0=0 $$
since by $t=x^2+y^2 \to 0$ by standard limits
$$(x^2+y^2) \log(x^2+y^2) =t\log t \to 0$$
and since $x^2+y^2 \ge 2xy$
$$0\le\frac{x^2y^2}{x^2+y^2} \le \frac... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Verification for a series of limits I ask kindly if my steps for these limits are right.
I start, always immediatly, checking the domain and this is an advice for my students of an high school.
I have, for example this function where the domain of
$$\operatorname{dom}\left(\frac{(1-x)^{\sqrt x}}{x-2}\right)=\{x\in \Bbb... | Let's not use L'Hopital's rule in $(A)$.
$$\bbox[yellow,5px,border:2px solid red]{\frac{(x-1)^{\sqrt x}}{x-2}=\frac{x^{(\sqrt x-1)}(1-x^{-1})^{x(\sqrt x)^{-1}}}{1-2x^{-1}}=P(x)\cdot Q(x)}\tag{1}$$
where, $$P(x)=x^{(\sqrt x-1)}, \quad \text{and} \quad Q(x)=\frac{(1-x^{-1})^{x(\sqrt x)^{-1}}}{1-2x^{-1}}$$
Note that $\li... | {
"language": "en",
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Weird looking ODE solution: How to verify this is indeed a solution? I will try to express my question using an example.
Consider this homogeneous ODE: $y' = \frac{y-x}{x+y }$
Its solution is: $\boxed{\frac12 \log\left( \frac{y^2(x)}{x^2} +1\right) - \log(x) + \arctan\left( \frac{y(x)}{x}\right) = c} \quad c\in \mathb... | Use the substitution $y = xv$
$$v+xv' = \frac{x(v-1)}{x(v+1)} \implies xv' = -\frac{v^2+1}{v+1}$$
which is separable
$$\int\frac{v}{v^2+1} + \frac{1}{v^2+1} \:dv = -\int\frac{dx}{x} \implies \frac{1}{2}\log\left(v^2+1\right)+ \tan^{-1}v = -\log|x| + C$$
which means the solution can be given as
$$\frac{1}{2}\log\left(x^... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Understanding the exactness of a sequence in Kummer Theory Here are the prerequisites and the parts which cause me trouble (taken from Milne's Fields and Galois Theory):
In particular, I don't quite understand the exactness at $H^1(G,\mu_n)$ and what it has to do with Hilbert 90 (as mentioned in the text).
If I unders... | Hilbert 90 has two common formulations -- one is the form you are quoting (where it is only valid if $E/F$ is a cyclic extension, by the way), and the other says $H^1(\text{Gal}(E/F), E^\times) = 0$ (any $E/F$ Galois -- this can be translated to your cyclic example by using an explicit description of the cohomology of ... | {
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Find a limit involving floor function I had to find the following limit: $\displaystyle{\lim_{x \to \infty}}\frac{x}{\lfloor x \rfloor}$ Where $x\in\mathbb{R}$ and $f(x)= {\lfloor x \rfloor}$ denotes the floor function.
This is what I did:
I wrote $x-1\leq \lfloor x \rfloor\leq x$. Then $\frac{1}{x}\leq\frac{1}{\lfloor... | It is correct. Slightly easier $\lfloor x\rfloor = x-\{x\}$ So $x/\lfloor x \rfloor= \frac{1}{1-\{x\}/x}$ so the limit is $1$ since $0\le\{x\}<1$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Question about group operation in Fundamental group I want to show that if the fundamental group $\pi_1(X,x_0)$ is abelian where $X$ is path-connected, then for any two paths $h_1,h_2: I\to X$ from $x_0$ to $x_1$, $\beta_{h_1}\equiv \beta_{h_2}$ where $\beta_h([f])=[h^{-1}*f*h]$. It seems that most of the solutions loo... |
The concatenation operation is, in general, is defined for two paths such that the initial point of one path coincides with the terminal point the other.
As you can see from the picture the paths has right identity and left identity as well as right and left inverses. Hence even though your [$h_1$] doesn't belong to t... | {
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Ideals with infinitely many generators As the title may suggest, I am not sure if I understand the ideals with infinitely many generators correctly. Let us take the ring $R=\mathbb{Z}[x_1,x_2,x_3,\dots]$ and consider the ideal $(x_1,x_2,x_3\dots)$ generated by all the indeterminates in the polynomial ring $R$. As a set... | Writing $$(x_1,x_2,x_3,\dots)=\{f_1x_1+f_2x_2+f_3x_3+\dots|f_1,f_2,f_3,\dots\in R\}$$ is rather imprecise, because (as you said), you cannot add infinitely many elements in a ring (unless you have a topology). It would be better to write instead
$$
(x_1,x_2,x_3,\dots) = \left\{\sum_{i\geq 1} f_i x_i\mid f_i\in R\textrm... | {
"language": "en",
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Let $X$ be any set $\tau : = \{X-\{a_1 , a_2 ,......, a_n\} \cup \phi\}$. Is $(X , \tau )$ $T_1$?
I can not understand the topology $\tau$. Can anyone please help me to understand?
| The topology that is meant is most likely:
$$\tau = \{X\setminus F\mid F\subseteq X\text{ finite }\} \cup \{\emptyset\}$$
And a topology on $X$ is $T_1$ iff for all $x \in X$, $\{x\}$ is closed. Now decide... Whether $X$ is a $T_2$ space will depend on the size of $X$..
| {
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"timestamp": "2023-03-29T00:00:00",
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How to show that $ C := A^3-3A^2+2A = 0$?
Let $$A = \begin{pmatrix} -1 & 1 & 2 \\ 0 & 2 & 0 \\ -1 & 1 & 2 \end{pmatrix}$$
Let $C:= A^3-3A^2+2A $.
Show that $C=0$.
I know that $A$ is diagonalisable with $\operatorname{spec}(A)=\{0,1,2\}$.
I have no clue how to approach that problem. Any advice?
| Here is a way to obtain the value of $p(A)$, where $p(X)= X^3+3X^2+2X$, using Hamilton-Cayley:
The characteristic polynomial of $A$ is $\:\chi_A=X(X-1)(X-2)=X^3-3^2+2X$, and we have
$p(X)=\chi_A(X)-6X^2$, so
$$p(A)=\chi_A(A)-6A^2=-6A^2$$
so you have only one matrix multiplication to do.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is this type of 'cfrac' known $\ldots\frac{a_n}{b_n+}\ldots\frac{a_2}{b_2+}\frac{a_1}{b_1+a_0}$? I have looked in a few well known books on cfracs but did not encounter anything related to the following quantity:
$$
\ldots\frac{a_n}{b_n+}\ldots\frac{a_2}{b_2+}\frac{a_1}{b_1+a_0}=\ldots\cfrac{a_n}{b_n+\cfrac{a_{n-1}}{b_... | Your problem is convergence.
Ordinary continued fractions are well-behaved objects because we can truncate them at various points, getting a sequence of rational numbers, which (at least assuming all coefficients are positive integers) converge to some limit. Extending the truncation has a smaller and smaller effect on... | {
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Number of non negative integer solutions of $x+y+2z=20$
The number of non negative integer solutions of $x+y+2z=20$ is
Finding coefficient of $x^{20}$ in
$$\begin{align}
&\left(x^0+x^1+\dots+x^{20}\right)^2\left(x^0+x^1+\dots+x^{10}\right)\\
=&\left(\frac{1-x^{21}}{1-x}\right)^2\left(\frac{1-x^{11}}{1-x}\right)\\
=&\... | Solving another way one has $0\le z \le10$ so we have in total the solutions of
$$x+y=20\space\space\text{ for } (x,y,0)\\x+y=18\space\space\text{ for } (x,y,1)\\x+y=16\space\space\text{ for } (x,y,2)\\x+y=14\space\space\text{ for } (x,y,3)\\x+y=12\space\space\text{ for } (x,y,4)\\x+y=10\space\space\text{ for } (x,y,5)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3813339",
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Is $\exists x(x=a)$ a valid formula in first order logic with equality? Under Tarskian semantics, is $\exists x(x=a)$ true in every interpretation in first order logic (FOL) with equality? If so, could $\exists x(x=a)$ be considered a tautology?
This question was motivated by another which was closed for lacking clarit... | Your question does not specify what $a$ is. The simpler case is if $a$ is a constant-symbol. Your posted answer is more or less correct for that case, but there are some issues. Firstly, it is not correct to write "$a≠a$" when referring to the element that $a$ is interpreted as in a structure $M$. Either write "$M ⊨ a≠... | {
"language": "en",
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"source": "stackexchange",
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How many $3$-digit numbers are there whose sum of digits is $10$? I did just counted all in gap of every $100$ numbers. Like in $100-200$, there are $10$ nos.($109,118,127, \ldots,181,190$). And then $9$ nos. in $200-300$, $8$ in $300-400$ ... and so on. By this I am getting $10+9+...3+2=54$, but answer given in India'... | This problem is equivalent to partitioning ten items to three boxes.
For example $433$ whose sum of digits is 10. $433$ can be represented as partition of items as below:
$$\circ \circ \circ \circ \mid \circ \circ \circ \mid \circ \circ \circ$$
More examples:
$109$
$$\circ \mid \mid \circ \circ \circ \circ \circ \c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3813775",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Consequence of uniform continuity $f:\mathbb R \rightarrow \mathbb R$ is a uniformly continuous function such that $f(n) \rightarrow 0$ as $n \rightarrow \infty$ where $n \in \mathbb N$. I have to find a counterexample to show that $f(x)$ may not converge to $0$ as $x \rightarrow \infty$. If we assume that $f(x^2)$ is ... | For easier notation let $g(x)=f(x^2)$. By assumption we know that $g(\sqrt{n})\to 0$ when $n\to\infty$. Now, we have:
$\sqrt{n+1}-\sqrt{n}=\frac{1}{\sqrt{n+1}+\sqrt{n}}\to 0$
So now let $\epsilon>0$. Since $g$ is uniformly continuous there is some $\delta>0$ such that $|x-y|<\delta$ implies $|g(x)-g(y)|<\frac{\epsilon}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3813925",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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T-annihilators and whether it kills the set Question : $V=F[t]v$ is T-cyclic, $g(t)$ is a polynomial. Show that if $g(T)v=0$, then $g(T)=0$.
Any ideas how to solve this?
plus, does this statement implies T-annihilators of v are exactly same as Ann(T)? I don't really get the differences between two concepts.
| Firstly, note $V$ being $T$-cyclic means $V = \text{span}\{v, T(v), T^2(v), \cdots \}$, and that $\{v, T(v), T^2(v), \cdots \}$ form a basis for $V$. The moment you are given $g(T)v=0$, you can conclude that $V$must be finite dimensional, since you may write $g(T)v= \sum_i g_i T^i v = 0$, so you have finitely many basi... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Can this monstrous expression be simplified? $\sqrt{\left( r_d \cos\left(\frac{-4(C - X) \csc(2α)}{Z}\right) + r_p \left(\frac{2C \tan(α) - 4 (C - X) \csc(2α))}{Z}\right) \sin\left(\frac{-4 (C - X) csc(2α)}{Z}\right) - m X \tan(α) \sin\left(\frac{-4 (C - X) \csc(2α)}{Z}\right)\right)^2 + \left( r_d \sin\left(\fra... | It's often much better to simplify as soon as possible. In this case, the parameterized $x$ and $y$ values at the specified value $\gamma_0 := -4(C-X)\csc(2\alpha)/Z$ reduces fairly nicely:
$$\begin{align}
x &=\tfrac12 mZ \left(\;\cos\gamma_0 + \gamma_0 \cos\alpha \sin(\alpha+\gamma_0)\;\right) \tag1\\[4pt]
y &=\tfrac1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3814259",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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$\int \frac{2x^7+3x^2}{x^{10}-2x^{5}+1}dx$
$$\int \frac{2x^7+3x^2}{x^{10}-2x^{5}+1}dx$$
I have no idea how to approach this problem.
I just know that I have to express the top expression in some form of a derivative of the expression in the denominator. But I just couldn't figure out how.
Any help would be appreciate... | Rewrite the expression as
$$\int\frac{2x^7+3x^2}{(x^5-1)^2}dx = \int \frac{2x+\frac{3}{x^4}}{\left(x^2-\frac{1}{x^3}\right)^2}dx = \frac{-1}{x^2-\frac{1}{x^3}}+C $$
by dividing top and bottom by $x^6$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Finding solution of $y' = \frac{(y^2 - 4yt + 6t^2)}{t^2}$ with initial condition $y(2) = 4$ Concerning the ordinary differential equation
$$y' = \frac{(y^2 - 4yt + 6t^2)}{t^2}$$
with initial condition $y(2) = 4$.
My text book gives the solution as $y = 2t$, and so does MATLAB.
However, I only see the following :
\begin... | You've shown
$$ tv' = v^2 - 5v + 6 = (v-2)(v-3).$$
and since $v = y/t$, the initial condition gives $v(2) =2$. Thus
$$\frac{dv}{(v-3)(v-2)} = \frac{dt}{t}$$
cannot be done since $v-2$ is at the bottom.
This is exactly the case where sometimes separation of variable does not work: indeed for a simple problem like
$$ y'... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Convergence of fixed points as a consequence of pointwise convergence? Setup: Let $p := p(n)$ be such that $\lim_{n \to \infty} pn = \lambda > 0$, and suppose $f_n(t) := (1 + (t-1)p)^n$, with domain $[0, 1]$.
Goal: I am trying to show that if $\theta_n$ is a fixed point of $f_n$:
$$
\theta_n = f_n(\theta_n), \quad \tex... | Your sandwich argument estimate is already essentially uniform. Namely,
$$e^{\lambda((t-1)+\epsilon)} - e^{\lambda((t-1)-\epsilon)} = e^{\lambda(t-1)} (e^{\lambda \epsilon} - e^{-\lambda \epsilon}).$$
As $\epsilon \to 0$, the right-hand-side becomes $1-1=0$.
Edit: Some more details. There's probably a cleaner way to ar... | {
"language": "en",
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What is the correct expression about Units digit? (revised) I want to express it accurately.
For example, consider $1872_{(9)}$. I would like to refer $2$.
There are many expression such as 'units of digit of $1872_{(9)}$ is 2', 'unit digit of $1872_{(9)}$ is 2' so on.
What is the correct expression?
Moreover, consider... | Taken from computer science, usually the $2$ in $432$ is referred to as the Least Significant digit, or LSd. and the $4$ would be the Most Significant digit, or MSd.
In any base, the right-most digit is the one with the least weight, thus is the least significant.
This also comes from the fact that a number in base $b... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Radius of the circumscribed circle of an isosceles triangle An isosceles triangle $ABC$ is given $(AC=BC).$ The perimeter of $\triangle ABC$ is $2p$, and the base angle is $\alpha.$ Find the radius of the circumscribed circle $R$.
$$R=\frac{p}{2\sin\alpha(1+\cos\alpha)}$$
Let $CD=2R.$ The triangle $BCD$ is a right t... | Another simple approach. Let $x=AC=BC$. Then
$$2p=AC+BC+2AH\\=2x+2x\cos\alpha$$
and
$$R=\frac 12 CD=\frac 12 \frac{BC}{ \sin \alpha} = \frac{x}{2 \sin \alpha}$$
Now you can complete the solution by a simple substitution.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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$\prod_{n=1}^\infty \frac{2n+1}{2n}$ diverges This is the Problem from the textbook "Intro to analysis" by Arthur Mattuck
Prove that $a_k := \prod_{n=1}^k \frac{2n+1}{2n}$ is strictly increasing and not bounded above.
Strict increasing is trivial: $\frac{a_{k+1}}{a_k} > 1$. But I am stuck at showing $a_k$ is not boun... | It is greater than $b_k=\prod_{n=1}^k\frac{2n+2}{2n+1}$ and $a_kb_k=k+1$ because the products telescope.
So $a_k\gt\sqrt{k+1}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3815230",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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About a subset of $\mathbb Q[x]$ of polynomials $f$ such that $f(n)=f(-n)$ for every $n$ in $\mathbb N$ Let $A=\{f \in \mathbb Q[x] : f(n)=f(-n)$, for every $n \in \mathbb N\}$.
Show that
*
*$A$ is a subring of $\mathbb Q[x]$.
*$A$ is a Euclidean Domain.
*For every $f \in A$ we have $f(r)=f(-r)$, for every $r \in \... | If $f(x)\in A$, let $g(x)=f(x)-f(-x)$. Then $(\forall n\in\Bbb N):g(n)=0$. Since $g(x)$ has infinitely many zeros, it is the null polynomial. Therefore, $(\forall r\in\Bbb Q):f(r)=f(-r)$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Example of a cancellative power semigroup Let $ \, S \, $ be a semigroup such that $ \ |S| \geq 2 \ $. Its power semigroup is the power set $ \, \wp(S) \, $ together with the binary operation
$$XY = \{ xy \in S : x \in X, \ y \in Y \} \ \ .$$
I am interested in the semigroup $ \ Q_S = \wp(S) \setminus \{ \varnothing \}... | Let $a \in S$ and let $a^+ = \{a^n \mid n > 0\}$ be the subsemigroup of $S$ generated by $a$. Since $aa^+ = a^+a^+$, one gets $a = a^+$ by right cancellation. Thus $a = a^2$ and $S$ is an idempotent semigroup. Furthermore, $aS = aaS$ implies $S = aS$ by left cancellation.
Consequently, $SS = \bigcup_{x \in S} xS = S = ... | {
"language": "en",
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Sets by Inclusion Clarification \begin{align}
A &= \lbrace 70, 210, 280\rbrace\\
B &= \mathbb{Z}\\
C &= \lbrace n\in\mathbb{N}\,|\, n=7 m \,\mbox{for some} \, m\in\mathbb{Q} \rbrace\\
D &= \lbrace n\in\mathbb{N}\,|\, n=35 m \,\mbox{for some} \, m\in\mathbb{N} \rbrace \\
E &= \lbrace n\in\mathbb{N}\,|\, n=7 m \,\mbox{fo... | $A$ by inspection and $C,D,E$ by definition are subsets of $\Bbb N$, and $\Bbb Z$ contains all of $\Bbb N$.
$C$ is in fact just equal to $\Bbb N$ as $n \in \Bbb N$ means $n = 7\cdot \frac{n}{7}$, so $7$ times a rational.
$A$ consists of multiples of $35$, so $A \subseteq D$.
A multiple of $35$ is always a multiple of $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3815674",
"timestamp": "2023-03-29T00:00:00",
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Controlling $C^1$ norm by Hölder norm and $C^0$ norm I've been trying to do this problem for a little while and have not made much progress. I want to show that for any $\epsilon>0$, there exists some $C_{\epsilon}>0$ such that
$$
\Vert u'\Vert_{L^{\infty}} \le \epsilon\sup_{x\neq y}\frac{|u'(x)-u'(y)|}{|x-y|^{\alpha}}... | Suppose
$$ |u'(x) - u'(y)| \le \frac1\epsilon |x-y|^\alpha \tag 1$$
Suppose $u'(x) \ge 1$. Then from (1), for $|y-x| \le (\frac12 \epsilon)^{1/\alpha}$, we have that $|u'(x) - u'(y)| \le \frac12$, and hence $u'(y) \ge \frac12$.
Hence if $|x-y| = (\frac12 \epsilon)^{1/\alpha}$, then
$$ |u(x) - u(y)| = \left|\int_x^y u'... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Implicit Differentiation of $x+y= \arctan(y)$? I'm trying to verify that $x+y = \arctan(y)$ satisfies this differential equation: $1+(y^2)+(y^2)y'= 0$. To do so, I tried to differentiate $x+y = \arctan(y)$ to get $y'$, but only got so far:
$$1 = \left(\frac{1}{1+y^2}y' -1 \right) y'.$$
Now I'm not sure how I can isolat... | You had differentiated correctly, but made an error when you re-arranged your equation:
$$ [ \ x \ + \ y \ ] \ ' \ \ = \ \ [ \ \arctan(y) \ ] \ ' \ \ \Rightarrow \ \ 1 \ + \ y' \ \ = \ \ \frac{1}{1 \ + \ y^2} · y' $$
$$ \Rightarrow \ \ 1 \ \ = \ \ \frac{1}{1 \ + \ y^2} · y' \ - \ y' \ \ \Rightarrow \ \ 1 \ \ = \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3815870",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Tu's Manifolds Problem 20.10 I am trying to compute the Lie derivative $\mathcal{L}_X\omega$ of the 2-form
$$\omega:= xdy\wedge dz + (z-y)dx\wedge dy$$
against the vector field
$$X:=-y \frac{\partial}{\partial x} + x \frac{\partial}{\partial y}$$
on the 2 sphere $S^2 \subset \mathbb{R}^3$. I believe that the flow $F$ ... | Here's what I got, although it's not 100% certain that I got the correct results. You should check your calculations and mine again and compare them.
So $$L_X \omega = d i_X \omega + i_X d \omega.$$
First, we calculate $$i_X \omega = \omega(X, \cdot) = x \cdot \begin{vmatrix} x & 0 \\ dy & dz \end{vmatrix} + (z-y)\cdot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3815957",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 0
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Spectral Theorem for unbounded self adjoint operators So I've been working on the Spectral Theorem for self adjoint unbounded Operators with the Book by Rudin and got to a problem:
Let $(X,\mathcal{A})$ be a measure space, $H$ a complex Hilbert space and $P:\mathcal{A}\rightarrow B(H)$ a resolution of the identity. The... | $|f_n(x)| \leq \|\Psi (f)\| $ for all $x$ outside some set $E_n$ of measure $0$. If $x \notin \cup_n E_n$ and $f_n(x) \to f(x)$ then $|f_n(x)| \leq \|\Psi (f)\| $ for all $n$ so $|f(x)| \leq \|\Psi (f)\| $. This proves that $\|f\|_{\infty} \leq \|\Psi (f)\| $.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Bondarenkos counter-example in dimension $\geq 65$ to Borsuk's conjecture. Just to remind you, Borsuk conjectured that:
Every subset $E\subset \mathbb{R}^d$ can be partitioned into $(d+1)$ sets with smaller diameter.
Even though this conjecture had been proven to be wrong, the search for the smallest dimension in whi... | First:
The considered paper by Bondarenko gave counterexamples for (all integer) dimensions from 65 onwards but not for a smaller dimension.
Second:
The mentioned arXiv reference is for the preprint. The final paper appeared in the journal Discrete & Computational Geometry, volume 51, issue 3 (April 2014). Meanwhile, i... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Locally constant group schemes Let $k$ be a field of characteristic $p$, then the group scheme $\mu_\ell$, if $p$ does not divide $\ell$, is étale-locally isomorphic to $\mathbb{Z}/\ell\mathbb{Z}$. I have two quick questions which I don't feel very certain. Is $\mu_p$ fppf locally isomorphic to $\mathbb{Z}/p\mathbb{Z} ... | I guess if $\mu_p$ were locally isomorphic to $\mathbb{Z}/p\mathbb{Z}$, there would be a $k$-algebra $A$ such that $\mu_{p,A}\simeq \mathbb{Z}/p\mathbb{Z}_A$. But for a field $K$ which is a $A$-algebra, $\mu_{p,A}(K)=1$ but $\mathbb{Z}/p\mathbb{Z}_A(K)$ is the abstract group $\mathbb{Z}/p\mathbb{Z}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3816400",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
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Computing card probabilities $1$ to $13$ and whose colors are red, green, blue, or yellow. You draw the topmost three cards. What's the probability that:
*
*All three cards are green or blue. I think the answer is ${26\choose 3}/{52\choose 3}$ since there are $26$ green or blue cards, and we pick three.
*Exactly t... | The first three and the last one are correct.
In the fourth one you’ve overcounted. Specifically, you’ve counted each set of $3$ that contains exactly two $12$s twice, once for each $12$, and each set of $3$ that includes exactly three $12$s three times, once for each $12$: any one of the $12$s can be the one that you ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3816553",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Question about using the notation "$\pm$" to represent the answer briefly. I saw this problem in the book with answer:
Solve the equation $z^4+z^2+1=0$.
The book gave the final answer as:
$$z_1=\frac{1}{2}+ \frac{\sqrt{3}}{2}i, \quad
z_2=\frac{-1}{2}- \frac{\sqrt{3}}{2}i,\quad
z_3=\frac{-1}{2}+ \frac{\sqrt{3}}{2}i,\q... | There are, I think, two questions implicit in the original post:
*
*Is it acceptable to write $\pm a \pm b$?
I think that the answer to this is a qualified "yes". As noted by Calum Gilhooley, there are notable texts and authors who have written $\pm a\pm b$ to denote a four element set, i.e.
$$ \pm a\pm b = \{ a+b, ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Integral involving composition of Trig. and Inverse Trig. functions. For a minor class i am taking this year, I found the following integral in a problem set, and where i had no luck in evaluating it: $$\int \cos(2\cot^{-1}\sqrt{\frac{(1-x)}{(1+x)}})dx$$.
I proceeded as follows:
->first, let $x=\cos(2\theta)$ $\implies... | Hint:
WLOG let $x=-\cos2t,0\le2t\le\pi$
$$\text{arccot}\sqrt{\dfrac{1-x}{1+x}}=\text{arccot}(\cot t)=t\text{ as } 0\le t\le\dfrac\pi2$$
$$\cos\left(\text{arccot}\sqrt{\dfrac{1-x}{1+x}}\right)=\cos(2t)=-x$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3816978",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 3
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Proving $(a+b+c) \Big(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\Big) \leqslant 25$ For $a,b,c \in \Big[\dfrac{1}{3},3\Big].$ Prove$:$
$$(a+b+c) \Big(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\Big) \leqslant 25.$$
Assume $a\equiv \text{mid}\{a,b,c\},$ we have$:$
$$25-(a+b+c) \Big(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\Big) =\dfra... | Let $f(a,b,c)=(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)$. Note that $f$ is concave of each variable (if other variables are fixed). Hence, since concave on $I$ fucntion attains its maximum at endpoint of $I$ (here $I=[m,M]=\left[\frac{1}{3},3\right]$)
$$
\max_{(a,b,c)\in I^3} f=\max_{(a,b,c)\in\{m,M\}^3} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3817104",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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negative variance example I came across the following matrix.
I notice, that it is symmetric and real and has an eigenvalue of -0.01091786.
This must mean, that there exists a vector $x \in \mathbb{R}^{10}$ for which it holds
$$ xVx^T<0$$
but I have no idea how to come about an example of such vector? - I especially w... | Here is one trick for finding eigenvectors associated with an eigenvalue. It is guaranteed to work if the characteristic polynomial is also the minimal polynomial, but may fail otherwise:
If $p$ is the characteristic polynomial of $A$, then by the Cayley-Hamilton theorem, $p(A) = 0$. If $\lambda$ is an eigenvalue of $A... | {
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Is it possible to define $f: S \rightarrow T$ such that $f$ is continuous and onto for each of the following pairs of $S$ and $T ?$ Is it possible to define $f: S \rightarrow T$ such that $f$ is continuous and onto for each of the following pairs of $S$ and $T ?$ For each pair, provide an example of one such $f,$ if po... | The answer to ii is yes. For example, we could consider the function $f:S \to T$ defined by
$$
f(x,y) = (\sin^2(2 \pi x),\sin^2(2 \pi y)).
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3817436",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving $6(x^3+y^3+z^3)^2 \leq (x^2+y^2+z^2)^3$, where $x+y+z=0$ Question :
Let $x,y,z$ be reals satisfying $x+y+z=0$. Prove the following inequality:$$6(x^3+y^3+z^3)^2 \leq (x^2+y^2+z^2)^3$$
My Attempts :
It’s obvious that $x,y,z$ are either one negative and two positive numbers or one positive and two negative num... | Scaling, one may assume that $z = -1$, so that $x + y = 1$. Then the desired inequality is
$$6(x^3 + y^3 - 1)^2 \leq (x^2 + y^2 + 1)^3$$
Since $x^3 + y^3 = (x + y)^3 - 3xy(x + y) = 1 - 3xy$ and $x^2 + y^2 + 1 = (x + y)^2 - 2xy + 1 = 2 - 2xy$, the desired inequality is
$$54(xy)^2 \leq (2 - 2xy)^3$$
Thus it makes sense t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3817541",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 4
} |
Mixed directional derivative of second order and the Hessian matrix First order directional derivative in 2 dimensional $xy$-coorinate system is
$$D_{\bf{u}}=u_1\frac{\partial}{\partial x}+u_2\frac{\partial}{\partial y}$$
I thought to go further and analyse second order directional derivatives.
The formula for general ... | I'm not sure if this is exactly what you're looking for, but if we write $\mathbf u=\begin{bmatrix} u_1\\u_2\end{bmatrix},\ \mathbf v=\begin{bmatrix} v_1\\v_2\end{bmatrix}$, then the formula becomes
$$D^2_{\bf{u}\bf{v}} = \mathbf v^T\mathbf H\mathbf u$$
Let us write $$\begin{align}
f_{11}&=\frac{\partial^2f}{\partial x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3817703",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
definite integral, regularized hypergeometric function I am currently stuck at the following equation: $\frac{\sqrt{m} \sec \left( (m+1)\pi \right)}{4\Gamma(m)} \int\limits_{0}^{\infty} \sqrt{z}\: {}_1\mathcal{M}_1\left(\frac{1}{2},\frac{3}{2}-m,\frac{-mz}{4} \right) dz=1$ $\forall$ $m \in \mathbb{Z}^+$, where ${}_1\ma... | To evaluate
\begin{equation}
I=\frac{2\Gamma(m-\frac{1}{2})}{m\pi\Gamma(m)}\int\limits_{0}^{\infty}t^{\frac{1}{2}}{}_1F_1\left(\frac{1}{2},\frac{3}{2}-m,-t
\right)\,dt
\end{equation}
we replace the hypergeometric function by its representation in terms of
Laguerre polynomials (see
here)
\begin{equation}
{}_1F_1(a, a - ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3817814",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
What is the partial sum $\sum_{k=1}^n (-1)^{k-1}\frac{\left(\frac{1}{9}\right)^k}{2k-1}$? I have a problem where, after some work, I've arrived at
$$6 \times \lim_{n \to ∞} \sum_{k=1}^n (-1)^{k-1}\frac{\left(\frac{1}{9}\right)^k}{2k-1}$$
and I need to find the partial sum $$\sum_{k=1}^n (-1)^{k-1}\frac{\left(\frac{1}{9... | Hint:
$$\arctan(x)=\sum_{k=0}^{\infty}(-1)^k\frac{x^{2k+1}}{2k+1}$$ if $|x|< 1.$ Now consider the following shift of the summation index $k\rightarrow k-1.$
\begin{align} \arctan(x)&=\sum_{k=1}^{\infty}(-1)^{k-1}\frac{x^{2k-1}}{2k-1}. \space \text{Now let $x=\frac{1}{3}$.} \end{align}
\begin{align} \text{Your fi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3817938",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Quotient rule for multivariable functions $\newcommand{\mbf}{\mathbf}$
Let $f,g:\mathbb R^n\to \mathbb R$ be differentiable at $a$.
$Df(a)$ is the unique linear transformation $\mathbb R^n\to\mathbb R$ such that
$$\lim_{\mbf h\to\mbf 0}\frac{|f(\mbf a+\mbf h)-\mbf f(\mbf a)-Df(\mbf a)(\mbf h)|}{|\mbf h|}=0.$$
I want t... | You only need to show that $D\frac{1}{g(a)}=-\frac{Dg(a)}{g^2(a)}$ and for this is necessary and sufficient that
$$\lim_{h\to 0}\frac{|1/g(a+h)- 1/g(a)+D(g(a))(h)/g^2(a)|}{|h|}=0.$$
Indeed, note that
$$\frac{|1/g(a+h)- 1/g(a)+D(g(a))(h)/g^2(a)|}{|h|}=\frac{|g^2(a)-g(a)g(a+h)+g(a+h)Dg(a)(h)|}{|g^2(a)g(a+h)h|}\leq\frac{|... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3818066",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
$\int_I f\,dm\geq\vert I\vert$ for any interval $I$, prove $f(x)\geq 1\text{ a.e.}$ Problem:
$f$ is Lebesgue integrable function defined on $[a,b]$, and $\int_I f\,dm\geq\vert I\vert$ for any interval $I\subset [a,b]$. Prove $f(x)\geq 1 \text{ a.e. } x\in [a,b]$
As open sets can be written as the union of countable d... |
Lebesgue's Theorem:--- Here one can use a lemma related Hardy-Littlewood maximal function
which says that for a locally Lebesgue integrable function $\varphi$ defined over
$\Bbb R^n$ we have $$\lim_{r\to
0+}\frac{1}{m(B_r(x))}\int_{B_r(x)}\varphi(y)\,dy=\varphi(x)\text{ a.e. } x\in
\Bbb R^n.$$
See Folland's Real Ana... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3818192",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
$\sum_{m=1}^{\infty}\sum_{n=1}^{\infty} \frac{m²n}{n3^m +m3^n}$ $\sum_{m=1}^{\infty}\sum_{n=1}^{\infty} \frac{m²n}{n3^m +m3^n}$.
I replaced m by n,n by m and sum both which gives term $\frac{mn(m+n)}{n3^m +m3^n}$.how to do further?
| A closely related summation doable by hand is when $3^m$ multiplies the denominator of OP:
$$S=\sum_{m=1}^{\infty}\sum_{n=1}^{\infty} \frac{m²n}{3^m(n3^m +m3^n)}~~~(1)$$ $$S=\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\frac{1}{a_m(a_m+a_n)},~~ a_k=3^k/k~~~(2)$$
Interchange $m$ and $n$, then
$$S=\sum_{n=1}^{\infty}\sum_{m=1}^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3818304",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Sum of 3 unit vectors being shorter than 1 What is the probability for the sum of three unit vectors to be shorter than 1?
The vectors' direction angles has uniform distribution on $[0, 2\pi]$.
I've made simulations and I also used Wolfram Alpha to solve the final equation below, so I'm pretty sure, that the result is
... | The first two points $z_+$, $z_-$ have an angle $\alpha'$ among themselves, where $\alpha'$ is uniformly distributed in $[0,\pi]$. We may assume them as $\cos\alpha\pm i \sin\alpha$, where $\alpha$ is uniformly distributed in $\bigl[0,{\pi\over2}\bigr]$. This gives $z_++z_-=2\cos\alpha=:z_*$. The unit circle with cente... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3818454",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 1
} |
Let the distribution of a random vector (X,Y) be given by density. Are X,Y independent? Let the distribution of a random vector $(X,Y)$ be given by density.
$$f(x,y)=\left\{\begin{matrix}
e^{-x-y} & x,y\geq 0\\
0 & x<0 \hspace{2mm} \vee y<0
\end{matrix}\right.$$
Are $X,Y$ independent?
My take:
So I have to calculate ... | You are mostly correct, but as Henry commented, you should include the support with the expressions (and using indicators to do so saves type space). This verifies that the support for the product of the marginal functions is identical to the support for the joint function as well as the product and the joint being e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3818584",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Are 2 norm and infinity norm equivalent in an infinite dimensional vector space space I am trying to answer the following question:
Are the 2-norm and the infinity norm equivalent in $l^2$, the space of real valued sequences which are square summable?
($l^2 := \big\{ x = (x_i)_{i \in \mathbb{N}} \ \ \big| \ \ \sum_{i=1... | Take $x_n = (1,1,1,...,\underset{n-th}{1}, 0,0,....)\in \ell^2.$ Then $$||x_n ||_{\infty} =1$$ but $$||x_n||_2 =\sqrt{n}$$ so these norms can't be equivalent on $\ell^2$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3818715",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
A mathcounts question The question is this.
There are $240$ pairs of numbers such as $(7,5318)$ or $(17,358)$ that can be formed using each of the digits $1$, $3$, $5$, $7$ and $8$ exactly once. What is the largest possible product of two such numbers?
Why wouldn't the answer be $87531 \times 87513$?
The answer key s... | No digits can be repeated, so 87531 * 87513 is not an option. Look for a two digit $*$ 3 digit number combination
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3819126",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Suppose Kn decomposes into edge-disjoint triangles. Prove that 6 | n − 1 or 6 | n − 3. My problem is finding a way to prove this. I was thinking that any Kn graph could have at least one triangle as long as n is 3 or higher. Other than that I do not know where to go with proving this is true.
| Hint: To show this, you want to show that $n$ is odd and that $n$ is not equivalent to $2\bmod 3$. It's good to do those separately.
For each part, it's useful to look at some examples. Why can't you split the edges of $K_4$ into three edge-disjoint triangles? How about $K_5$? Try counting things, like degrees or the t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3819382",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
A bartender stole champagne from a bottle that contained 50% of
A bartender stole champagne from a bottle that contained 50% of spirit and he replaced what he had stolen with champagne having 20% spirit. The bottle then contained only 25% spirit. How much of the bottle did he steal?
My approach: Let the total quantit... | Without (much) math.
Must steal enough champagne so that weighted average goes 5/6 of the way from 50 to 20
(i.e. $[25-20] \times 5 = [50 - 25]$).
Therefore, he must steal 5/6 of the bottle.
I mention this approach not as a recommended approach for someone new to the problem, but rather as a way of developing intuitio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3819503",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Distribution of first $9$ natural numbers The question states
The first $9$ natural numbers are to be divided in three groups $g_1$, $g_2$ and $g_3$ of equal size. In how many different ways can this be done if the sum of numbers in each group is odd?"
What I thought was to split it into cases. The three numbers chos... | The first nine natural numbers contain exactly five odd numbers. Each group must contain an odd number of natural numbers if the sum of the numbers in each group is to be odd. The only way to do this is to have one group with three odd numbers and two groups with exactly one odd number.
Choose which three odd numbers... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3819652",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Ambient isotopy and isotopy In the lecture note of my knot theory course, there is a very short explanation why we need ambient isotopy instead of just isotopy. It uses the following figure to illustrate without any explanation. I don't understand why it's OK to make a knot become unknot when it's isotopy, but not OK w... | The idea of the diagram is to demonstrate that isotopy between two embeddings is an incorrect notion of knot equivalence. Note that a knot is an embedding:
$k:S^1\to \mathbb{R}^3$ (or more conveniently $\mathbb{S^3}$)
Given two knots (embeddings) $k_0$ and $k_1$, we may construct an isotopy of embeddings:
$k_t:S^1\time... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3819811",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 1,
"answer_id": 0
} |
Simplify $\sum_{i=1}^{n}\sum_{j=i+1}^{n} (i+j)$ $$\sum_{i=1}^n \sum_{j=i+1}^n (i+j)$$
I'm pretty sure it's somewhat simple to solve but I can't get it done.
I know that the given sum is equal to $\frac{n}{2} (n^2-1)$.
I've been on this for 3h+ now and I'm really hoping someone can give me a hint :)
| Here is to help you to start:
\begin{align}
\sum_{i=1}^n \sum_{j=i+1}^n (i+j) &= \left( \sum_{i=1}^n \sum_{j=i+1}^n i\right)+\left( \sum_{i=1}^n \sum_{j=i+1}^n j\right) \\
&= \left( \sum_{i=1}^n i (n-(i+1)+1)\right)+\left( \sum_{i=1}^n \left(\sum_{j=1}^n j -\sum_{j=1}^i j \right)\right) \\
\end{align}
Try to simplify i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3819902",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Proving two equivalent statement that come from Otto Holder's theorem $a \in \ell_q$ and $x \in \ell_p$. We also have $1/p + 1/q = 1$.
I want to show that
$$
\cfrac{|a_ix_i|}{||x||_p ||a||_q}
\leq
\cfrac{1}{p}\left(\cfrac{|x_i|}{||x||_p} \right)
+\cfrac{1}{q}\left(\cfrac{|a_i|}{||a||_q} \right) \\
\implies
\cfrac{\su... | Hint:
Use Young's inequality
for each summand.
This will look similar to your assumed inequality (maybe you meant to add the exponents so that it becomes Young's inequality).
edit: To address the edit of the question: Yes, there does seem to be a mistake. In the second part the exponents $p$ and $q$ are missing over th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3820118",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Cauchy-Schwarz Inequality problems Let $a,$ $b,$ $c,$ $d,$ $e,$ $f$ be nonnegative real numbers.
(a) Prove that
$$(a^2 + b^2)^2 (c^4 + d^4)(e^4 + f^4) \ge (ace + bdf)^4.$$
(b) Prove that
$$(a^2 + b^2)(c^2 + d^2)(e^2 + f^2) \ge (ace + bdf)^2.$$
I'm not sure how I should start approaching both problems. I believe I shoul... | These are both CS inequality applications, you should try yourself. Here is the first one:
*
*$(c^4+d^4)(e^4+f^4) \geqslant (c^2e^2+d^2f^2)^2$
*$(a^2+b^2)(c^2e^2+d^2f^2) \geqslant (ace+bdf)^2$
Now combine the two to get what you want.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3820233",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
If $a^2+b^2-ab=c^2$ for positive $a$, $b$, $c$, then show that $(a-c)(b-c)\leq0$
Let $a$, $b$, $c$ be positive numbers. If $a^2+b^2-ab=c^2$. Show that
$$(a-c)(b-c)\leq0$$
I have managed to get the equation to $(a-b)^2=c^2-ab$, but I haven't been able to make any progress.
Can someone help me?
| By the law of sines, we have that
$$a^{2}+ b^{2}- 2ab\cdot\sin 60^{\circ}= c^{2}\Rightarrow c:={\rm med}\left \{ a, b, c \right \}\Rightarrow \left ( a- c \right )\left ( b- c \right )\leq 0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3820511",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 3
} |
A differentiable function on $(a,b)$ with nonzero derivative over $(a,b)$ where $f'(c)>0$ for some $c\in(a,b)$ means $f'(x)>0$ for all $x\in(a,b)$. It certainly seems correct that if you have at least one point $c\in(a,b)$ such that $f'(c)>0$, where $f'(x)\ne0$ for all $x\in(a,b)$, then $f'(x)>0$ for all $x\in(a,b)$. A... | According to Darboux's Thereom, if $f$ is a differentiable function in an interval $I$, then the range of its derivative function $f'$ must still be an interval. Using the given condition, we can easily get the conclusion.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3820623",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
When are eight integers entirely determined by their pairwise sums? Alice picks 8 numbers, which are not necessarily different. Once she has picked them, she writes out the addition of all the pairs on a piece of paper, which she gives to Basil. Basil wins if he can guess correctly the original n numbers, which Alice c... | Hint: if the collections $(a_1, \dots, a_k)$ and $(b_1, \dots, b_k)$ have identical pairwise sums, then the collections $(a_1, \dots, a_k, b_1+m, \dots, b_k+m)$ and $(b_1, \dots, b_k, a_1+m, \dots, a_k+m)$ also have identical pairwise sums. (The number $m$ has to be such that $a_i \neq b_j \pm m$ for all $i,j$, so that... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3820737",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 2,
"answer_id": 0
} |
Does this inequality hold true for all $\alpha\in\mathbb{R}$? Let $x$ and $y$ be two real and positive numbers. Let $\alpha\in\mathbb{R}$. I am trying to understand if the inequality
$$ x^{\alpha} + y^{\alpha} \leq (x+y)^{\alpha}$$
holds true. By attemps, I found that it holds true only if $\alpha \geq1$. Could anyone ... | If the inequality holds for all $x, y \geq 0$ the take $x=y=1$ to get $2 \leq 2^{\alpha}$ which implies $\alpha \geq 1$.
If $\alpha \geq 1$ consider the function $f(x)=(x+y)^{\alpha}-x^{\alpha}-y^{\alpha}$ for fixed $y$. Since $f'(x)=\alpha (x+y)^{\alpha -1} -\alpha x^{\alpha -1} \geq 0$ the function is increasing on ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3820921",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
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