Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Is there an (interesting) mathematical theory in first order logic that is inconsistent with Peano Arithmetic? There are first order theories that don't entail PA, like Tarski's elementary geometry.
Is there one that isn't consistent with PA, aka T + NON-PA would be sound? (Ideally it wouldn't be a "pathological" exam... | There's always PA+$\neg$Con(PA), but that's a bit pathological. More seriously, the most obvious one to my mind is the theory of the field of real numbers, $Th(\mathbb{R};+,\times)$. Perhaps surprisingly, even though the reals are bigger, their theory is simpler - Tarski showed that it's decidable!. The field of ration... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2433155",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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No projective modules in category. I have an exercise that reads:
Let $C$ be a the category of all finite $\mathbb{Z}$-modules, prove that there are no projective modules in $C$.
So, in order for $P$ to not be projective $\mathbb{Z}$-module I must prove that for every surjection $g: P \to M$ and every $f: N \to M$ it c... | Since $\mathbb{Z}$ is a PID, projective modules $P$ are free $\mathbb{Z}$-modules. However, since $P$ is finite of order $n$, we have $nP=0$, so that $P$ is not free - see this MSE-question.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2433254",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What is the smallest number of integer weights required to exactly balance every integer between $1$ and $40$?
What is the smallest number of integer weights required to exactly balance every integer between $1$ and $40$.
I do not really understand what this problem is asking for me to do. Any suggestions are appreci... | Suppose you have a set of weights consisting of specific integral weights, and you have a balance. The idea is that someone could put any integral weight up to $40$ on the left hand side, and you get to use your weights on the right hand side.
Suppose your set of weights consist of one weight each of $1,2,5,10,20$. The... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2433372",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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How do I find Jordan basis? I have a matrix:
$$A=\begin{pmatrix}0&1&0\\-4&4&0\\-2&1&2\end{pmatrix}$$
solving $\det|A-\lambda{I}|$ I got characteristic polynom that equals to $(2-\lambda)^3 = 0$ for eigenvalue found two eigenvectors and one generalized eigenvector: $v_1=(1,2,0)\quad v_2=(0,0,1) \quad v_3=(1,0,0)$
What d... | Here is the way to go: consider the sequence of kernels:
$$\{\,0\,\}\varsubsetneq\ker(A-2I)\varsubsetneq\ker(A-2I)^2\subset\dots$$
The sequence stops after step $2$ since
$$A-2I=\begin{bmatrix}-2&1&0\\-4&2&0\\-2&1&0\end{bmatrix}\qquad (A-2I)^2=\begin{bmatrix}
0&0&0\\0&0&0\\0&0&0\end{bmatrix}$$
$A-2I$ has rank $1$, henc... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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How to apply bayes rule for this problem? Calculate P(X=T|Y=F) given P(Y=F|X=T) = 0.3, P(X=T) = 0.5, and P(X=F,Y=F)=0.2. Note that T represents true, F represents false, all variables are binary.
| Use Bayes:
$$P(X=T|Y=F)=\frac{P(Y=F|X=T)P(X=T)}{P(Y=F)}$$
And for denominator:
$$
\begin{align}
P(Y=F)&=P(Y=F,X=T)+P(Y=F,X=F)\\
&=P(Y=F|X=T)P(X=T)+P(Y=F,X=F)
\end{align}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2433818",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Solving for $\log^*n$ I know the iterative logarithm can only produce 1 of 6 numbers. However, I don't really understand how to solve. Can someone please explain how to solve $\log^*n$ where $n$ is any number, lets say like 100. Would there be any difference for $\lg^*n$?
| Suppose we'd like to solve $\log^*100$ with a base-10 logarithm. $\log100=2$, which is greater than 1, but $\log\log100$ is less than 1. Thus $\log^*100=2$.
For $\lg^*100$ (base 2) we have
$$\lg100=6.644$$
$$\lg\lg100=2.732$$
$$\lg\lg\lg100=1.450$$
$$\lg\lg\lg\lg100=0.536$$
so $\lg^*100=4$.
These functions sometimes co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2433949",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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Proof - raising adjacency matrix to $n$-th power gives $n$-length walks between two vertices I came across the formula to find the number of walks of length $n$ between two vertices by raising the adjacency matrix of their graph to the $n$-th power.
I took me quite some time to understand why it actually works. I thou... | Your idea looks correct. Just some remarks on your notation.
You chose $P(n)$ to denote the induction statement. And you (apparently) chose $P_{i,j}^{(n)}$ to denote the number of $v_i$-$v_j$-walks of length $n$. But you use it in the following inconsistent ways: $P(n)$, $P_{i,j}$, $P_{i,j}(n)$. The first one is the mo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2434064",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 1,
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} |
A formula for $1^4+2^4+...+n^4$ I know that
$$\sum^n_{i=1}i^2=\frac{1}{6}n(n+1)(2n+1)$$
and
$$\sum^n_{i=1}i^3=\left(\sum^n_{i=1}i\right)^2.$$
Here is the question: is there a formula for
$$\sum^n_{i=1}i^4.$$
| yes there is a formula, $$\sum_{i=1}^ni^4=1/30\,n \left( 2\,n+1 \right) \left( n+1 \right) \left( 3\,{n}^{2}+3
\,n-1 \right)
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2434311",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Find mistake in solving $\sin 2x=\sin x + \cos x$ I am solving $\sin 2x = \sin x +\cos x $ for $0\le x \le 360$
$$\sin 2x = \sin x +\cos x$$
$$2 \sin x \cos x =\sin x + \cos x$$
$$(\cos x + \sin x) ^{2} - (\sin x)^{2} - (\cos x)^{2} =(\cos x + \sin x)^{2} - 1=\sin x + \cos x$$
Let $\cos x + \sin x = y$
$$y^{2} - 1= y$$... | Consider these three equations:
\begin{align}
\sin 2x = \sin x + \cos x \tag{1} \\
(\sin x + \cos x)^2 - 1 = \sin x + \cos x \tag{2} \\
(\sin 2x)^2 -1 = \sin 2x \tag{3}
\end{align}
You were correct in determining that (1) and (2) are equivalent. Also, (1) does imply (3).
However, what you failed to notice is that (3) d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2434399",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
Prove $\left(2017^{2018}+2017^{2017}\right)^{2018}>\left(2018^{2017}+2017^{2017}\right)^{2017}$ Can someone give me a hint for this problem (for secondary student):
Prove that: $$\left(2017^{2018}+2017^{2017}\right)^{2018}>\left(2018^{2017}+2017^{2017}\right)^{2017}$$
P/s: I've thinking about using the fact that $n... | Hint:
$$
\frac{2017^{2018}}{2017^{2017}}=2017
$$
but
$$
\begin{align}
\frac{2018^{2017}}{2017^{2017}}
&=\left(1+\frac1{2017}\right)^{2017}\\
&\le\left(1+\frac1{2017}\right)^{2018}\\
&\le\left(1+\frac1{2016}\right)^{2017}\\
&\qquad\ \ \vdots\\
&\le\left(1+\frac11\right)^2\\[6pt]
&=4
\end{align}
$$
To show the last serie... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2434651",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 0
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How to find $\lim_{(x,y)\rightarrow(0,0)} \frac{x^2\sin^2(y)}{x^2+3y^2}$ Find the following $$\lim_{(x,y)\rightarrow(0,0)} \frac{x^2\sin^2(y)}{x^2+3y^2}$$
How to solve this limit? I have tried to use polar system, like sub everything with $x=rcos\theta$ and $y=\sin\theta$, but r cancel each other.
| I think it means
$$0< \frac{x^2\sin^2y}{x^2+3y^2}< \frac{x^2y^2}{x^2+3y^2}<x^2$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2434761",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
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Solve $\frac{1}{\sin^2{(\arctan{x})}}-\frac{1}{\tan^2{(\arcsin{x})}}=4x^2.$ I just need help to finetune this solution. Only having a correct answer is not sufficient to comb home 5/5 points on a problem like this. Thoughts on improvements on stringency? Is there any logical fallacy or ambiguity? Any input is very welc... | The allowed values for $x$ are in $[-1,1]$, because of $\arcsin x$, but $0$ should also be excluded. Also $-1$ and $1$ must be excluded because of $\tan\arcsin x$.
(Note: your $x\in[-\pi/4,\pi/4]$ is wrong and the probable cause for the low grade.)
The equation remains the same if we change $x$ into $-x$, so we can lim... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2435018",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 0
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Solving a differential equation, second pair of eyes needed. I have a simple ODE which contains a constant k. Solving this ODE gives a solution containing k. If I set $k=0$ in the solution, I do not get the solution that I get if I set $k=0$ in the original ODE. I have been through my workings over and over, but can... | If $k=0$ then $T'=0\iff T=cst$
If $k\neq 0$ then homegeneous equation is $T'=kT\iff T=Ce^{kx}$
Constant variation to find a particular solution with RHS :
$\require{cancel}T'=\cancel{Cke^{kx}}+C'e^{kx}=k(T-T_0)=\cancel{kCe^{kx}}-k(Ax+B)\iff C'=-k(Ax+B)e^{-kx}$
$C=(Ax+B+\frac Ak)e^{-kx}+D$
$T=Ax+B+\frac Ak+De^{kx}$
I ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2435124",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Automorphisms of $\mathbb P^1_{\mathbb k}$ over a algebraically closed field $\mathbb k$.
Let's consider an algebraically closed field $\mathbb k$. Consider the birational map $\phi: \mathbb P^1_{\mathbb k} \to \mathbb P^1_{\mathbb k}$. How do I show that if there are $f,g$, homogeneous poynomials, such that $$\phi[x:... | Let me just summarize what was discussed on the comment section.
A birational map $\phi$ is given by a rational function $z \to \frac{f(z)}{g(z)}$ with $f,g$ polynomials. The inverse is also given by a rational function. By this fact we obtain that $z \to \frac{f(z)}{g(z)}$ is automorphism of $\mathbb k(z)$ and the au... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2435252",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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Find the limit of $ \frac{\cos (3x) - 1}{ \sin (2x) \tan (3x) } $ without L'Hospital technique I would like to find the limit as $x$ goes to zero for the following function, but without L'Hospital technique.
$$ f(x) = \frac{ \cos (3x) - 1 }{ \sin (2x) \tan (3x)} $$
This limit will go to zero (I had tried using calcul... | The thinking/attempt in the question post is quite complicated. Simple idea related to @ParamanandSingh 's comment. Multiply the function by $ \frac{\cos(3x)+1}{\cos(3x)+1} $, we get
$$ f(x) = \frac{\cos^{2}(3x) - 1}{\sin(2x) \tan(3x)(\cos(3x)+1)} =\frac{- \sin(3x) \cos(3x)}{\sin(2x) (\cos(3x)+1)} = \frac{- (2x) \sin(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2435551",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 5
} |
Loops when drawing constantly changing angles and lines. I start by drawing a line of 1 unit on the $x$ axis. I turn left (from the perspective of an ant on the line) by an angle of $\alpha$ and I draw a second segment of length $u$ from my endpoint of the first segment. I then turn another angle $\alpha$ and then draw... | The loci are polygons containing start point on the circle as we know the sum of external angles $=360^0$ for a full closed regular polygon. In general it would be infinite sided irregular polygon. The "polygon" can be drawn even on BASIC.
Diameter $D$ of approximate circum-circle so formed $ =\dfrac{1+u}{2 \alpha} $, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2435649",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What is the probability of getting the sum $26$ when $7$ chips are taken out? Suppose you have a bag in which there are $10$ chips numbered $0$ to $9$. You take out a chip at random, note its number and then put it back. This process is done $7$ times and after that the numbers are added.
What is the probability that... | The number of ways of drawing 7 chits with sum equal to 26 is the coefficient of $x^{26}$ in the expansion
\begin{align*}
(1+x+x^2+\cdots +x^9)^7 &= \left(\frac{1-x^{10}}{1-x}\right)^7 \\
&=(1-7x^{10} + 21x^{20} - \cdots)\left(1+\binom{7}{1}x + \binom{8}{2}x^2 + \cdots\right)
\end{align*}
Coefficient of $x^{26}$ is
$$\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2435798",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What is wrong in my attempt to rotate two orthogonal vectors? Let $V=(1,0,0)$ and $W=(0,1,0)$ be two vector. Consider the following transformation:
$\pi/4$ rotation around the $z$-axis $V$ and $W$ concurrent counterclockwise, then $\pi/4$ rotating the result concurrent in direction from $y$-axis to $z$-axis (around $x$... | If the only transformations you perform are rotations around the origin, leaving the origin fixed, it is impossible to transform the pair of vectors
$(1,0,0)$ and $(0,1,0)$
into the pair of vectors $(1,1,1)$ and $(-1,1,1).$
In fact, you start out with two vectors in the $x,y$ plane;
after a rotation around the $z$ axi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2435874",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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To prove $\lim_{x\to 0}\frac{e^x-e^{x\cos x}}{x-\sin x}=3$ I came across this question to prove the given limit
$$\lim_{x\to 0}\frac{e^x-e^{x\cos x}}{x-\sin x}=3$$
First I tried using LHospital's rule directly.
Then I tried using expansion of $e^x$ and then using LHospital's rule but I am getting stuck.
| Use $\cos(x) = 1 - \frac12 x^2 + \cal{O}(x^4)$ and $\sin(x) = x - \frac{x^3}{6} + \cal{O}(x^5)$. Then
$$
\lim_{x\to 0}\frac{e^x-e^{x\cos x}}{x-\sin x}=\lim_{x\to 0}e^x\frac{1-e^{-x^3/2 + \cal{O}(x^5)}}{x-\sin x}= \lim_{x\to 0}\frac{{x^3/2 + \cal{O}(x^5)}}{x^3/6 + \cal{O}(x^5)}=\lim_{x\to 0}\frac{{1/2 + \cal{O}(x^2)}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2436145",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
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Group action on a Cartesian product Let $G$ be a finite group acting on a finite set $X$. Then naturally $G$ acts on $X \times X$ by $g.(x,y)=(g.x,g.y)$. Is there any way to find the number of orbits of the action of $G$ on $X\times X$ using the action $g$ on $X$? Are they related?
| (Not a complete answer, but just an instance of the particular case mentioned in the comments.)
Let's call "$\star$" the induced action of $G$ on $X\times X$. Then, for $\bar x:=(x_1,x_2)\in X\times X$, the pointwise stabilizer reads:
\begin{alignat}{1}
\operatorname{Stab}_\star(\bar x) &= \{g\in G\mid g\star\bar x=\ba... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2436231",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Polynomial Transformation Conventions Is there a particular reason why systems of linear equations are compiled as rows in a coefficient matrix, whereas polynomial transformations are compiled as columns in a coefficient matrix. Even then, why do both look at pivots in columns to determine linear independence, shouldn'... | In the context of linear algebra there are two possible conventions to transform linear equations into the framework of matrices resp. linear mappings.
(1) You consider vectors $x\in\mathbb{R}^d$ as columns, then solving the system of equations $0= \sum_{j=1}^d a_{ij}x_j$ (i=1,...,k) amounts to finding a column vector... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What is the rank in each of the following case? The columns of $A$ are $n$ vectors from $R^m$. If they are linearly independent, what is the rank of $A$? If they span $R^m$, what is the rank? If they are a basis for $R^m$ what then?
Here's my explanation:
The $n$ vectors are linearly independent, so $rank(A)=n$.
Now co... | Yes, you are correct. Note that if the columns of $A$ are $n$ linerly independent vectors of $\mathbb{R}^m$ than $n\le m$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2436495",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Help clarifying the steps to find the derivative of $y=(3x+1)^3(2x+5)^{-4}$ For the problem $y=(3x+1)^3(2x+5)^{-4}$ do I use the chain, quotient and product rules? If so how do I know what parts to break up and where the rules apply? For instance would I consider $f(x)$ to be $(3x+1)^3$ and $g(x)$ to be $(2x+5)^{-4}$... | You can chain the rules together. Each application of a rule gives you a simpler object to take the derivative of. I would start by using the product rule, giving $$\frac {dy}{dx}=(2x+5)^{-4}\frac d{dx}\left((3x+1)^3\right)+(3x+1)^{3}\frac d{dx}\left((2x+5)^{-4}\right)$$
Now you can use the chain rule on each term, g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2436600",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
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Repeated Roots of Polynomials Whose Coefficients are Either 0 or 1 Consider a polynomial of the form: $$f\left(z\right)=1+z^{p_{0}}+z^{p_{1}}+\ldots+z^{p_{N}}$$ where the $p_{n}$s are distinct positive integers. Are the roots of $f$ (in $\mathbb{C}$) necessarily simple (i.e., must they all have multiplicities of $1$)?
| No. We can construct as follows: The polynomial $(1+z^a)(1+z^b)$ will have two factors of $1+z$ if $a$ and $b$ are both odd. So if we choose $b$ a good bit bigger than $a$, the cross terms will miss each other and all coefficients will be $1$. For instance,
$$f(z) = (1+z^3)(1+z^5) = 1+z^3+z^5+z^8$$
has $-1$ as a dou... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2436708",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 2,
"answer_id": 1
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Nested Geometric Series Formula I would like to derive a general formula for series of the following type:
$$\sum_{n=1}^{\infty}\left(\left(\frac{1}{A^n}\right) + \sum_{j=n}^{\infty}\left(\frac{1}{A^n\cdot B^j}\right)\right)$$
I attempted first to decompose it into parts by considering the nested loop:
$$n=1$$
$$\frac{... | Let's ignore the outer summation for now and first just simplify the inner sum.
\begin{align}
\sum_{j=n}^{\infty} \frac{1}{A^n B^j} = \frac{1}{A^n}\sum_{j=n}^{\infty}\frac{1}{B^j} = \frac{1}{A^n B^n}\sum_{j=0}^\infty\frac{1}{B^j} = \frac{1}{A^n B^n}\frac{1}{1-1/B}
\end{align}
Notice that the power of $A$ doesn't depend... | {
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"timestamp": "2023-03-29T00:00:00",
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Prove $ x^{\frac{1}{1-x}} < \frac{1}{e} $ for $ 0 \leq x < 1 $ How can I prove the following statement?
If $ x \in \mathbb{R} $ and $ 0 \leq x < 1 $, then $$ x^{\frac{1}{1-x}} < \frac{1}{e}. $$
I could prove this statement: $ \lim\limits_{x \to 1} x^{\frac{1}{1-x}} = \frac{1}{e} $. I see that as $ x $ approaches $ 1 ... | We need to prove that
$$\frac{\ln{x}}{1-x}<-1$$ or
$$x-1-\ln{x}>0.$$
Let $f(x)=x-1-\ln{x}$.
Thus, $$f'(x)=1-\frac{1}{x}=\frac{x-1}{x}<0$$
and since $\lim\limits_{x\rightarrow1}f(x)=0$, we are done!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2436926",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Find a basis for $V$ for which it is the dual basis Let $V=(\Bbb R^3)$ and define $f_1, f_2, f_3 \in V^*$ as follows:
$$f_1(x, y, z) = x − 2y,f_2(x, y, z) = x + y + z, f_3(x, y, z) = y − 3z.$$
Prove that $\{f_1, f_2, f_3\}$ is a basis for $V^*$, and then find a basis for $V$ for which it is the dual basis.
My work... | The first dual basis vector should be $(a, b, c) \in \mathbb{R}^3$ such that $f_1(a, b, c) = 1$, and $f_2(a, b, c) = f_3(a, b, c) = 0$. Write this system out to get
$$\begin{matrix}
a & -&2b & && = & 1 \\
a & +&b & +&c & = & 0 \\
& &b & -&3c & = & 0
\end{matrix}
$$
And solve. Now do the same for the second and third du... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find the value of $c$ to make $f_{XY}(x,y)$ a valid joint pdf $f_{XY}(x,y) = cx $ where $x>0$, $y>0$ and $2<x+y<3$
My approach:
\begin{align*}
1 &= \int_0^\infty\int_0^\infty cx\,dx\,dy
\\ &= \int_0^2\int_{2-x}^{3-x}cx \,dx\,dy
\end{align*}
I just want to know if the limits of the integrations are correct or not?
| The limits are not correct. Normally, it helps a lot drawing the domain you are dealing with:
Note that $x$ can vary from $0$ to $3$. Once you have that, then you can compute the limits for $y$, based on the lines $x+y=3$ and $x+y=2$
By the way, you cannot write
$$1 = \int_0^\infty\int_0^\infty cxdxdy,$$
since that in... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Evaluating limits without L'Hopital's I need to evaluate without using L'Hopital's Rule. They say you don't appreciate something until you lose it. Turns out to be true.
*
*$\lim_{x \to 0} \frac{\sqrt[3]{1+x}-1}{x}$
*$\lim_{x \to 0} \frac{\cos 3x - \cos x}{x^2}$
I have tried to "rationalise" Q1 by using the identi... | Q1 - good approach. But take $a=\sqrt[3]{1+x}$, $b=1$. See what happens. Not $\frac{0}{0}$.
Q2 - similarly - something should cancel and $\frac{0}{0}$ should vanish.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2437235",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 3
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The derivative of a discrete distribution function is zero a.e.? Suppose $\{q_n\}$ is an arbitrary enumeration of all rationals, and $$F(x)=\sum_{n=1}^{\infty}\frac1{2^n}\mathbf 1_{\{q_n\le x\}}$$ Then $F$ is a discrete distribution function with countable jump discontinuities (that are in $\mathbb Q$), but is continu... | Not necessarily. Assume for example that $$\pi<q_{2n}<\pi+5^{-n}$$ for every $n$ (note that enumerations with this property do exist since the subsequence $(q_{2n+1})$ is free), then $$F(\pi+5^{-n})>F(\pi)+4^{-n}$$ hence $$\frac{F(\pi+5^{-n})-F(\pi)}{5^{-n}}>\left(\frac54\right)^n$$ which implies $$\lim_{x\to0,x>0}\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2437465",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove of equality of function and composition $V, W$ are finite dimensional $\mathbb{K}$-Vectorspaces. $\,\,f: V \rightarrow W$ is an Isomorphism. $\,\,q_v \in \text{Quad}(V), \,\,q_w \in \text{Quad}(W)$.
It is known, that:
$$
q_v(v_1 + v_2) - q_v(v_1) - q_v(v_2) = (q_w \circ f)(v_1+v_2) - (q_w \circ f)(v_1) - (q_w \c... | Try setting $v_2 = v_1 = v$. Then your equality says:
$$
\begin {align}
q_V(2v) - 2q_V(v) &= q_W(f(2v)) - 2q_W(f(v)) \\
2q_V(v) &= 2q_W(f(v))
\end {align}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2437560",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Can a sequence contain an undefined term (e.g. a member acquired by dividing a number by zero)? I learned about the theorem that every convergent series is bounded.
However, take $a_n = \frac{1}{n-3}$ with $n=3$.
It is unbounded, since $\sup{a_n} = +\infty$ (EDIT: it is not.).
And yet, it is convergent with $\lim {a_... | A convergent sequence of real numbers is bounded. It's important to recognize that $\infty$ and $-\infty$ are not real numbers.
There are extensions of the real numbers in which these symbols become legitimate numbers, although even in those systems there are some restrictions on the operations that can be performed us... | {
"language": "en",
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"source": "stackexchange",
"question_score": "2",
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Estimating the standard deviation by simply looking at a histogram I would like to make a quick, rough estimate of what a standard deviation is.
Suppose i have the following histogram
By simply looking at it, I can say that the mean is around 10 or 9.8 (middle value) which, when calculating from my dataset, is actuall... | Assume normal distribution where 99.7% (~100%) of values fall within 3 standard deviations from the mean. This implies your $x_{min}$ and $x_{max}$ values define the full span of the domain and are each roughly 3 standard deviations from the mean, leading to:
$$ \sigma = \frac{x_{max} - x_{min}}{6} $$
In above case, $\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What is a natural number? According to page 25 of the book A First Course in Real Analysis, an inductive set is a set of real numbers such that $0$ is in the set and for every real number $x$ in the set, $x + 1$ is also in the set and a natural number is a real number that every inductive set contains. The problem with... | Defining the natural numbers (to within isomorphism) as a distinguished subset of a complete ordered field (COF) seems mathematically slick, but is unsatisfactory in two ways. First, definition should go from the simple to the complex and from the elementary to the advanced—not the other way round. Second, in order for... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 1
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Probability of different people being born on the same day of the week What is the chance that at least two people were born on the same day of the week if there are 3 people in the room?
I know how to get the answer which is 19/49 when considering all 3 people not being born on the same day. However, when I try to cal... | The only mistake you have committed is that you have not included the combinations while calculation of 2 were born on the same day
If you name the persons A, B ,C
Since the two of them may be
A B,
B C,
C A
Therefore you must also multiply it by a factor of 3
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Partial differential equations with related functions Suppose we have $f(x, y) \geq 0$ and $g(x, y) \leq 0$ continuous, such that $\frac{d^2f}{dx^2} = \frac{d^2g}{dy^2}$. Can you show that $\frac{df}{dx} = 0$ or provide a counterexample?
EDIT: $f, g: \mathbb{R}^2 \rightarrow \mathbb{R}$
| By chosing the domain $\mathbb{R}_+ \times \mathbb{R}_+$ and setting:
$$
f(x,y)=xy \geq 0\\
g(x,y)=-xy \leq 0
$$
Then we have
$$
f_{xx}=0=g_{yy}
$$
But
$$
f_x=y \neq 0
$$
If you want an answer for $\mathbb{R}^2$ as the "whole space", just specifiy it in your question. But this answer just came to my mind while readin... | {
"language": "en",
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"source": "stackexchange",
"question_score": "1",
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Isomorphism between groups defined by set builder notation How is it possible to show that for a set $G = \{x + y\sqrt{3}| x, y\in\mathbb{Z}\}$ and a set $H = \{2^i*5^j| i,j\in\mathbb{Z}\}$ that the group $[A, +]$ is isomorphic to $[B, *]$ (let $+$ be standard addition and $*$ be standard multiplication between reals)?... | Yes, because both groups are isomorphic to $(\mathbb Z \times \mathbb Z, +)$.
This gives an isomorphism $G \to \mathbb Z \times \mathbb Z \to H$:
$$
x + y\sqrt{3} \mapsto (x, y) \mapsto 2^x 5^y
$$
To see that this is a homomorphism, compute:
$$
(x + y\sqrt{3})+(x' + y'\sqrt{3})
=(x + x') + (y+y')\sqrt{3}
\mapsto 2^{x+x... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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determine the number of r-combinations of the multiset Determine the number of r-combinations of the multiset $\{1 \cdot a_1, \infty. \cdot a_2, ... , \infty \cdot a_k \}$.
The answer in the back of the book is $\binom{r+k-2}{k-2}+\binom{r+k-3}{k-2}$.
I see where $\binom{r+k-2}{k-2}$ comes from because it is the numb... | Let $x_j$ be the number of occurrences of $a_j$, where $1 \leq j \leq k$. Then we need to find the number of solutions of the equation
$$x_1 + x_2 + x_3 + \ldots + x_k = r \tag{1}$$
in the nonnegative integers subject to the restriction that $x_1 \leq 1$.
If $x_1 = 0$, equation 1 reduces to
$$x_2 + x_3 + \ldots + ... | {
"language": "en",
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can I find the closest rational to any given real, if I assume that the denominator is not larger than some fixed n Given $n\in\mathbb{N}$ and $r\in\mathbb{R}$(or $r\in\mathbb{Q}$), is that possible to find a rational $\frac{a}{b}$such that, $b<n$ and $\frac{a}{b}$ is the "closest" rational to r?
With "closest" I mean... | If you are given a real number, indeed the rationals "closest to it" on both the left and right will exist, and they will be very easy to find.
For example, let us consider $n=10$ and the real number $\pi$. We want to find the closest rationals to $\pi$ with denominator less than $10$. For this, we do something very s... | {
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Inequality on tangent and secant function
Let $\{\alpha, \beta, \gamma, \delta\} \subset \left (0,\frac {\pi}{2}\right)$ and $ \alpha + \beta+\gamma+\delta = {\pi}$. Prove that $\sqrt {2} \left(\tan \alpha +\tan \beta +\tan\gamma+\tan \delta\right)\ge \sec \alpha +\sec \beta +\sec \gamma +\sec \delta$
| let $\tan\alpha =a, \tan\beta =b, \tan\gamma =c, \tan\delta =d\implies a,b,c,d>0$
Also $\alpha+\beta+\gamma+\delta=180^{\circ}\implies \alpha+\beta=180^{\circ}-(\gamma+\delta)$
$\implies \tan (\alpha+\beta)=-\tan (\gamma+\delta)$
$\implies \frac{a+b}{1-ab}=-\frac{c+d}{1-cd}$
$\implies a+b+c+d=abc+abd+acd+bcd$ ___(i)
... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the area of a circle part A circle is centred at $(a,b)$ and has radius $d$. The value $k$ is such that the lines $y =k $ and $x=k$ both intersect the circle twice. Moreover, $a,b<k$, so that the point $(k,k)$ is inside the circle, but also above, and to the right of, the point $(a,b)$. This setup is illustrated b... | You have 6 areas
$P_1=\frac{d^2\pi}{4}$
$P_2=\arcsin(\frac{k-a}d)\frac{d^2}{2}$
$P_3=\arcsin(\frac{k-b}d)\frac{d^2}{2}$
$P_4=\cos\arcsin(\frac{k-a}d)\frac{d(k-a)}{2}$
$P_5=\cos\arcsin(\frac{k-b}d)\frac{d(k-b)}{2}$
$P_6=(k-a)(k-b)$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
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How to show $((a,b),b)=(a,b)$? I want to prove $((a,b),b)=(a,b)$ for $a$, $b$ integers with some formal mathematical statement like Bezout's identity i.e., $(a,b)=d=am+bn$ or something else.
Any ideas?
Here $(a,b)$ means gcd.
My first sight is let $(a,b)=d$ and factors $a=d q_a$, $b=d q_b$ and proceed.
I want more ... | Hint $\,\ ((a,b),b) = (a,(b,b)) = (a,b)\ $ by the gcd Associative Law.
Or directly $\,\ d\mid((a,b),b)\iff d\mid(a,b),b\iff d\mid a,b,\, b\iff d\mid a,b\iff d\mid (a,b)\,$ where we used the gcd Universal Property $3$ times.
Or $\ d\mid b\,\Rightarrow\, (d,b) = (d, b\bmod d) = (d,0) = d\,$ by the Euclidean algorithm, ... | {
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How to show $\sqrt{\log(1+x)}\leq 1+\log (x^2)$ for $x\geq 1$? Throughout the post, define $\log(x):=\log_e(x)$. Plotting the graph of the function
$$
f(x)=1+\log (x^2)-\sqrt{\log(1+x)},\quad x\geq 1,
$$
one can expect that $f(x)\geq 0$ for all $x\geq 1$.
I'm looking for a proof of it.
If one looks at its deriv... | [This is essentially Michael Rozenberg's answer. I would like to rephrase it a bit.]
It is mentioned in OP that it suffices to show that $f'(x)\geq 0$ for $x\geq 1$ since $f(1)>0$. On the other hand, when $x\geq 1$,
$$
4(1+x)\sqrt{\ln(1+x)}-x\geq4(1+x)\sqrt{\ln2}-x>(4\sqrt{\ln 2}-1)x
>0,
$$
which implies that $f'(x)>0... | {
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Expected value of the max of $2$ dice What is the expected value of the max of two dice?
I just wonder if there's a better way to get the answer to this question than listing out all possible outcomes and determining the expected value from there as this is actually an interview question?
Thanks
| Reasoning as for the time allowed to answer to an interview:
a) in a square $6 \times 6$, the number of cases with max no greater than $m$, will be the square $m \times m$.
b) so $m^2/36$ gives the CDF.
c) the median will be at $CDF = 1/2$ that is for a square of area $=18$, i.e. about $4.2$.
d) to be a bit more preci... | {
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Proof verification on countability of algebraic numbers? I'm writing a proof that the set of all algebraic numbers for a homework assignment, do all my steps look correct?
For each $i = 0, 1, 2,\ldots$ let $P_i := \{\text{all integer polynomials of degree $i$}\} := \{p_0 + p_1x + \cdots + p_i x^i \ : \ p_0,\ldots,p_i \... | This isn't quite right. You have the right idea: leveraging the countability of the integers. However, as you presented it, the set $P_k \cup \mathbb{Z} = \mathbb{Z}$, and I don't think this is what you want.
For instance, you give $P_0 = \mathbb{Z}$. Then you claim that $P_1 = P_0 \cup \mathbb{Z}$. This means that $P_... | {
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"timestamp": "2023-03-29T00:00:00",
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Does there exist an uncountable set $A \subset \mathbb{R}$, s.t for every $a \in A$ & every $\epsilon>0$, $(a-\epsilon,a+\epsilon)\not\subset A$?
Does there exist an uncountable set $A \subset \mathbb{R}$, such that for every $a \in A$ and every $\epsilon>0$, $(a-\epsilon,a+\epsilon)\not\subset A$?
I am not sure what... | Hint: The set of rational numbers is dense and countable.
| {
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"timestamp": "2023-03-29T00:00:00",
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Prove that $x^2 + 3 > 2x + 1$ for all values of $x$ how would I go about proving this:
$x^2 + 3 > 2x + 1$
I know that I have to complete the square but do I bring everything to one side and convert it to an equation or do I simply complete the first side?
Sorry if this seems nooby
| With Analysis:
Just to show there isn't a single way to prove inequalities.
The tangent to the parabola with equation $y=x^2+3$ at point $A=(1,4)$ has equation
$y=2(x-1)+4=2x+2$.
Now the function is convex, i.e. its graph is above each of its tangents, so for all $x$,
$$x^2+3\ge 2x+2~(>2x+1).$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 3
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Show that $x^2 = \sin (x)$ has exactly one positive solution I want to show that $x^2 = \sin(x)$ has exactly one positive solution.
We know that $x^2 - \sin(x)$ only has roots in the segment $[-1,1]$ and we also know that $x=0$ is a root.
How do I show that there exists exactly one root on $(0,\infty)$?
| hint
$f $ is continuous and differentiable at $(0,+\infty). $
Assume that there exist $x_1>x_2>0$ such that
$$f (x_1)=f (x_2)=0=f (0) $$
then by Double Rolle Theorem,
$$\exists c_1, c_2 \;\;:\;\; f'(c_1)=f'(c_2)=0$$
and
$$\exists c\in (c_1,c_2) \;\;:\;\; f''(c)=0$$
but
$$f''(x)=2+\sin(x)>0$$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Collection of subgroups The question revolves around a statement like the following:
Let $G_\lambda \ (\lambda \in \Lambda)$ be subgroups of $G$...
How do I collect subgroups of $G$ in a mathematically precise manner? Do I interpret $G_\lambda \ (\lambda\in \Lambda)$ as a function
\begin{align}
\lambda \mapsto G_\la... | It is usually referred to as a family which is exactly the same thing as a function.
Its codomain is $\{G_\lambda, \lambda\in \Lambda\}$, which is a set according to the axiom scheme of replacement. Alternatively, you could say its codomain is $\mathcal{P}(G)$, the powerset of $G$, or if you want the image $\{H\in \mat... | {
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Determine whether $ f(x,y)$ exists given the partial derivatives Find a function $z=f(x,y)$ whose partial derivatives are as given, or explain why this is impossible.
We have that $ f_x$ = $ 3x^2y^2-2x$, and $f_y$ = $ 2x^3y+6y$. where $ f_z$ denotes the partial derivative of the function $ f$ with respect to some vari... | You can answer that without any deeper analysis.
From $f_x=3x^2y^2-2x$ it can be seen that $y^2x^3-x^2+C_1+g(y)$ has to be derivatived with respect to $x$ to obtain $f_x=3x^2y^2-2x$.
From $f_y=2x^3y+6y$ it can be seen that $x^3y^2+3y^2+h(x)+C_2$ has to be derivatived with respect to $y$ to obtain $f_y=2x^3y+6y$.
Of cou... | {
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"source": "stackexchange",
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Prove that a conjugate of a subgroup is a subgroup
Let $G$ be a group and $H$ be a subgroup of $G$ and $a\in G$ fixed, then
$$H^{a}=aHa^{-1}=\{aHa^{-1} \colon h\in H\}$$
is a subgroup of $G$.
my attempt
Identity
$aha^{-1} \in H^a$
$aea^{-1} \in H^a\ \ \ \ $ Since $e ∈ H$
$aa^{-1} \in H^a$
Therefore $aa^{-1} = e ∈ H^{... | Let $H \leq G$
Let $H'= \{aHa^- \}$ be some congjugate Subgroup cnjugated by some $ a \in G$ where
$\{ah_1a^- , ah_2a^- .. \} \in H' $ for every $\{h_1 , h_2 ..\} \in H $
*
*$(ah_1a^-)(ah_2a^-) = ah_1h_2a^- = ah_3a^- \in H'$. This proves closure
*Associativity can be inhertied from group compsotion since they are ... | {
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Show that if $\sum \limits_{n=1}^\infty a_n$ converges then $\lim\limits_{n\to\infty}\frac{1}{n}\sum _{k=1}^n ka_k=0$ Show that if $\sum \limits_{n=1}^\infty a_n$ converges then $\lim\limits_{n\to\infty}\frac{1}{n}\sum \limits_{k=1}^n ka_k=0$
since given that $\sum\limits _{n=1}^\infty a_n$ converges
then we can say ... | $$\sum \limits_{n=1}^\infty a_n<\infty$$ converges then
for $\varepsilon >0$ there is $n_0$ such that
$$\left|\sum \limits_{n=n_0}^\infty a_n\right|<\varepsilon$$
let $n\ge n_0$ then,
$$\left|\frac{1}{n}\sum \limits_{k=1}^{n}ka_k\right| \le\left|\frac{1}{n}\sum \limits_{k=1}^{n_0} ka_k\right|+\left|\frac{1}{n}\sum \l... | {
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"answer_id": 1
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Limit of convergent series equal zero
Show that if $\sum_{n=1}^{\infty} a_n$ is convergent then
$$
\lim_{n\to\infty} \frac{1}{n} \sum_{k=1}^n k a_k=0
$$
If we let the limit of series be $A$ then it can be shown $\lim_{n\to\infty} (a_1+\dots + a_n)/n=A$ as well. But how to deal with the term $ka_k$?
| Let $A_n=\sum_{k=1}^n a_k\to A$ then, by summation by parts,
$$\sum_{k=1}^{n} ka_k=\sum_{k=1}^{n} k(A_k-A_{k-1})=\sum_{k=1}^{n} kA_k-\sum_{k=1}^{n}(k-1)A_{k-1}-\sum_{k=1}^{n}A_{k-1}=nA_n-\sum_{k=1}^{n}A_{k-1}$$
Hence, by the Stolz-Cesaro theorem,
$$\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^{n} ka_k=\lim_{n\to\infty}A_n
-\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2440023",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Equation of the bisector of the angle between two lines containing the given point
The general form of the equation of the angle bisector of two lines :
$$ \begin{align}L_1 &=a_1x+b_1y+ c_1=0
\\L_2 &= a_2x+b_2y+ c_2=0 \end{align}$$
Given as:
$$\dfrac{a_1x+b_1y+c_1}{\sqrt{a_1^2+b_1^2}}= \pm\dfrac{a_2x+b_2y+c_2}{\sqrt{a... | There is a much more elegant and geometric view point for deriving this equation. Let the lines be $L_1 = a_1 x + b_1 y + c_1=0$ and $L_2 = a_2 x + b_2 y + c_2=0$, the unit normal for these lines are given as :
$$ \begin{align} n_1 &= \frac{\nabla L_1}{|\nabla L_1|} \\ n_2 &= \frac{|\nabla L_2|}{|\nabla L_2|} \end{alig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2440168",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
If H and K are two normal subgroups of a group G .If the order of H and the order of K are relatively prime.prove that hk=kh My answer is : let $h= gxg^{-1}$ in $H$ for $x$ in $H $
and $k= gyg^{-1}$ in $K$ for $y$ in $K$
$hk =(gxg^{-1})(gyg^{-1})=g (xy) g^{-1}$
But $xy$ in $HK$. hence $HK$ is normal in $G$ i.e . $a... | My favorite proof of this:
Consider $hkh^{-1}k^{-1}$: since $H \lhd G$, we have:
$hkh^{-1}k^{-1} = h(kh^{-1}k^{-1}) \in H$. Since $K \lhd G$, we have:
$hkh^{-1}k^{-1} = (hkh^{-1})k^{-1} \in K$.
Therefore $hkh^{-1}k^{-1} \in H \cap K$, and by Lagrange, the order of $hkh^{-1}k^{-1}$ divides $\gcd(|H|,|K|) = 1$.
Hence $hk... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2440258",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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General Cesaro summation with weight
Assume that $a_n\to \ell $ is a convergent sequence of complex numbers and $\{\lambda_n\}$ is a sequence of positive real numbers such that $\sum\limits_{k=0}^{\infty}\lambda_k = \infty$
Then, show that,
$$\lim_{n\to\infty} \frac{1}{\sum_\limits{k=0}^{n}\lambda_k} \sum_\limits{k=... | Let $\varepsilon >0$ and $N$such that $|a_k-l|\le \varepsilon $ for all $k>N$
Then, for $n>N$ we have,
\begin{split}\left| \frac{\sum_\limits{k=0}^{n}\lambda_k a_k}{\sum_\limits{k=0}^{n}\lambda_k} -l\right|
&= &\left| \frac{\sum_\limits{k=0}^{n}\lambda_k (a_k - l)}{\sum_\limits{k=0}^{n}\lambda_k} \right|\\
&= &\left| ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2440333",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Why is choosing from the remainder incorrect in solving this combination question? There are 4 badminton, 5 volleyball and 6 swimmers. What are the number of ways to form a delegation of 4 players which has to have at least 1 player from each of the 3 sports.
My answer was $\binom{4}{1}\cdot\binom{5}{1}\cdot\binom{6}{1... | You are double counting some teams. Consider two volley ball players $V_1$ and $V_2$. You can choose $V_1$ as part of 5C1 and $V_2$ as part of 12C1. Again $V_2$ can be selected as part of 5C1 and $V_1$ as part of 12C1. This counts this combination twice.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2440435",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
If $x, y \in X$ with $x \neq y$, then there exists $f \in X^*$ such that $f(x) \neq f(y)$. Let $X$ be a normed linear space. Prove that if $x, y \in X$ with $x \neq y$, then there exists $f \in X^*$ such that $f(x) \neq f(y)$.
Here $X^*$ denotes the dual space of $X$.
I am getting some smell of using Hahn Banach theore... | Let $x,y \in X$ such that $x \neq y$.
Assume that $f(x)=f(y) ,\forall f \in X^*\Rightarrow f(x-y)=0$
From the consequences of Hahn-Banach exists $f_0 \in X^*$ such that $||f_0||=1$ and $f_0(y-x)=||x-y||$.
But $$f_0(x-y)=0 \Rightarrow ||x-y||=0 \Rightarrow x=y$$
contradicting our hypothesis that $x \neq y$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2440520",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 5,
"answer_id": 0
} |
For $A$ an infinite set, and $f: A^A \to A$, does there exist $a\in A$ such that $f^{-1}(\{a\}) \cong A^A$? I've been wondering about this question, and unfortunately I do not have much experience with set theory.
The notation: $A^A := \{g:A\to A\}$.
The reason I suspect this is very weak actually, and that is that for... | Yes.
König's theorem states that the cofinality of $2^\kappa$ is strictly greater than $\kappa$. This means that if we partition $2^\kappa$ into at most $\kappa$ parts, one of them is necessarily of size $2^\kappa$.
To see why, note that you can formalize cofinality by stating that the cofinality of an infinite cardin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2440622",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is every abstract category a concrete category of structures? In conceptual mathematics by Lawvere and Schanuel, it says:
There are many more categories than just those given by abstract types of structure; however, those can be construed as full subcategories of the latter, so that the notion of map does not change.
... |
there are categories without a faithful functor to Set
There are in fact many such categories. https://arxiv.org/abs/1704.00303
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2440692",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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In what kind of space does this object live? Let me quickly build up some background.
One way to build a hypercube is to take cubes, and start gluing them together, face to face, such that each edge is shared by $3$ cubes. You complete the hypercube with $8$ cubes. This involves rotating cubes in $4$ dimensions, but i... | I have a more straightforward answer which doesn't require any calculations.
These tori you have there are actually Clifford tori which live in 4D. Look at the vertices, each of them has degree 4 and they are connected each to each. You have to take into account the wrapped edges also. And all of these edges are the sa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2440778",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
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"answer_id": 2
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Proving that $\sqrt{2}$ is irrational with a math level of a middle school student? So I have a friend, whose professor challenged the class to prove that $\sqrt{2}$ is irrational by using only middle school math level.
No one managed to do it, and neither did I (although I didn't give it much thought, as I have other ... | Hopefully, the following argument (adapted from Cauchy)) might be at the middle school level, as it uses only the notions of integer and decimal parts of a number.
We prove that if $x=\sqrt 2$ is rational, it must necessarily be an integer. As it is clear no perfect square is equal to $2$, it will prove that $x$ is irr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2440864",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 4,
"answer_id": 2
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Using the Chinese Remainder theorem to solve the systems of congruence's. We got introduced to the Chinese Reminder Theorem, but I still haven't quite grasped it and I have a problem that asks:
Find $x \bmod 77$ if $x \equiv 2 \pmod 7$ and $x \equiv 4 \pmod {11}$.
My attempt was like so:
$$B = 7 * 77 * 11 = 5929$$
... | I don't know if this is what you have on your mind, but I would do it like this:
$x=7a+2$ and $x= 11b+4$ for some integers $a,b$ so $7a= 11b+2$ and thus $7|11b+2$.
Multiply this with 2 and we get $7|22b+4$ so $7|b-3$. Now we can write $b=7t+3$ and thus $x=77t+37$. So $x \equiv 37 \pmod {77}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2441003",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
} |
How to find the base r? $\sqrt{144_r} = 12_r$
What is r?
The method I used is:
$\sqrt{ ((1 × r^2) + (4 × r^1) + (4 × r^0))} = ((1 × r^1) + (2 × r^0))$
and I tried solving this equation but I got now where.
the solution to this question according to the book is $r\geq 5$.
| Simplify the expression first (remember that $r^0 =1$):
$$\sqrt{r^2+4r+4}_r = (r+2)_r$$
$$\sqrt{(r+2)^2}_r = (r+2)_r$$
$$(r+2)_r = (r+2)_r$$
What's wrong with this statement:
"This equality is true when $r ≥ -2$, because the square root of a real number is imaginary. Therefore, it is true for all (positive) bases."?
Hi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2441150",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finding sum of the series $\sum_{r=1}^{n}\frac{1}{(r)(r+d)(r+2d)(r+3d)}$ Find the sum: $$\sum_{r=1}^{n}\frac{1}{(r)(r+d)(r+2d)(r+3d)}$$
My method:
I tried to split it into partial fractions like: $\dfrac{A}{r}, \dfrac{B}{r+d}$ etc. Using this method, we have 4 equations in $A,B,C,D$! And solving them takes much time. I... | HINT: $$\frac{1}{r(r+d)(r+2d)(r+3d)} = \frac{1}{2d^2}\left(\frac{1}{r(r+3d)}-\frac{1}{(r+d)(r+2d)}\right) = \frac{1}{6d^3}\left(\frac{1}{r}-\frac{1}{r+3d}\right)-\frac{1}{2d^3}\left(\frac{1}{r+d}-\frac{1}{r+2d}\right).$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2441296",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 3
} |
Optimization problem - Graphically How can we draw the cobb-douglas optimization problem with constraint and solve it graphically?
We have the function $U(x,y)=x^{0.4}y^{0.3}$.
Do we have to draw the constraint and some level curves of $U(x,y)$ ?
|
Do we have to draw the constraint and some level curves of U(x,y) ?
That is right. I would recommend to use a function plotter since a single utility function only is hard enough to draw. You solve the utility function for y:
$$y=\left( \frac{\overline U}{x^{0.4}} \right)^{10/3}$$
For different levels of $\overline U... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2441426",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Try to calculate arithmetic mean of this periodic function I'm trying to do a arithmetic mean of this periodic function:
I choose this subintervals:
$$f(t)= \left\{ \begin{array}{lcc}
\frac{-2A}{t_0} t & if & \frac{-t_0}{2} \leq t < 0\\
\frac{2A}{t_0}t & if & 0 \leq t < \frac{-t_0}{2}
... | Just do it very carefully:
\begin{align}\overline{f}(t) &= \frac{1}{t_0}\int_{\frac{-t_0}{2}}^\frac{t_0}{2} f(t)\,dt\\
&= \frac{1}{t_0}\left(\int_{\frac{-t_0}{2}}^0 \frac{-2A}{t_0}t\,dt + \int_{0}^{\frac{t_0}{2}} \frac{2A}{t_0}t\,dt\right) \\
&= \frac{1}{t_0}\left(\left(-\frac{A}{t_0}t^2\right)\Bigg|^0_{-\frac{t_0}{2}}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2441528",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Limits and algebraic simplification Please let me know what is wrong in the following algebraic simplification which gives a wrong limit value.
$$\lim_{x \to \infty} (\sqrt {x^2 - 4x} - x)=\lim_{x \to \infty} (\sqrt {x^2 (1 - 4/x)} - x) $$
$$= \lim_{x \to \infty} (x\sqrt {1 - 4/x} - x)= \lim_{x \to \infty} x(\sqrt {1 -... | TL;DR: "When $x$ tends to infinity, $4/x$ will be negligible and hence [...]" is not justified algebraic simplification.
We could just as well say: "When $x$ tends to infinity $\sqrt{1-\frac 4x}-1$ is negligible..." That said, just replace $\sqrt{1-\frac 4x}-1$ with any $f(x)$ such that $\lim_{x\to\infty} f(x) = 0$.
W... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2441646",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 7,
"answer_id": 6
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Combinatorial Proof of $\binom{i+j}{j}S(n, i+j) = \sum_{k=0}^{n}\binom{n}{k}S(k,i)S(n-k,j)$ I would like to prove the following statements:
$\binom{i+j}{j}S(n, i+j) = \sum_{k=0}^{n}\binom{n}{k}S(k,i)S(n-k,j)$
combinatorially.
(where $\;S(i,j),\;i,j\in \Bbb Z\;$ denotes the Stirling number of the second kind, put labell... | HINT: Show that this is the number of ways to put $n$ labelled objects into $i$ red boxes and $j$ blue boxes. For the full combinatorial argument, hover below:
We can count these in two ways: first, just put $n$ objects into $i + j$ uncolored boxes, and then choose which boxes will be colored blue. This corresponds... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Let $(X,d)$ a metric space and $A \subset X$. $A$ is bounded if and only if $\operatorname{diam}(A)$ is finite. Good morning, I have a problem to prove the following theorem.
Theorem:Let $(X,d)$ a metric space and $A \subset X$. $A$ is bounded if and only if $\operatorname{diam}(A)$ is finite.
Prove:
$->$ A is bound... | Just do a direct proof:
Let $\operatorname{diam}(A)$ be finite, and define $R = \operatorname{diam}(A)+1 > 0$. Pick $a \in A$.
Then take any $x \in A$, then $d(x,a) \le \operatorname{diam}(A) < R$ (as $x,a \in A$ they are bounded by the $\sup$ of all distances between points of $A$, i.e. $\operatorname{diam}(A)$). Thi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2441906",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Convergence of $a_{n+1}=x+\sin a_{n}$ Assume $x \in \Bbb R$ is a real number, consider the following sequence:
$a_{n+1}=x+\sin a_{n}, a_0=0.$
For which $x$, the sequence above is convergent $?$ If $x \in \pi \Bbb Z$ then it's obvious, but I wonder what will happen for other $x$.
| Suppose that $f(t)=x+\sin(t)$ and $a=f(a)$. According to fixed point theorems, your sequence converges if
$$|f'(a)|<1$$
Which trivially holds if
$$a\notin\pi\Bbb Z$$
But if $a\in\pi\Bbb Z$, well, as you say, the convergence is obvious.
Thus, it converges for any $x\in\Bbb R$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2442003",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Laplace's equation, theoretical doubt I tried to solve Laplace equation in two dimensions
$$u_{xx} + u_{yy}=0$$
without any condition added. Just tried to solve it by separation of variables, so I assumed that the solution be of the form $u(x,y) = \Phi_x(x) \cdot \Phi_y(y)$.
Replacing the correct derivatives in the equ... | You have the Laplace equation
$$\partial_{x}^{2}\Psi(x, y)+\partial_{y}^{2}\Psi(x, y)=0$$
By letting
$$\xi=x+iy$$
$$\bar{\xi}=x-iy$$
The equation becomes
$$\partial_{\xi\bar{\xi}}^{2}\Psi(\xi, \bar{\xi})=0$$
Integrating with respect to $\xi$ we get
$$\partial_{\bar{\xi}}\Psi(\bar{\xi}, \xi)=f(\bar{\xi})$$
Integrating ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2442064",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Proof that $\sqrt {n+1} - \sqrt n ≤ 1$ I have some difficulties proving the following statement: $\sqrt {n+1} - \sqrt n ≤ 1$
Thus far I have the following:
The root of a positive integer is always smaller than or equal to the integer, since $a^2 ≥ a$ where a is a positive integer.
Therefore $\sqrt {n+1} ≤ \sqrt n + 1$ ... | 0) For $n=0:$ $(1)^{1/2}-0 =1$.
1) Consider $n \ne 0$.
$f(x) = x^{1/2}$ is continuos in $[n,n+1]$, differentiable in $(n,n+1)$.
Mean value Theorem:
$\dfrac{f(n+1) -f(n)}{1} = f'(t)$,
where $t \in (n,n+1)$.
$(n+1)^{1/2} - (n)^{1/2} =$
$ (1/2)(t)^{-1/2} \lt 1$,
where $t \in (n,n+1)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2442158",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 7,
"answer_id": 5
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Let $C$ be the Cantor set and $\mu$ be the uniform measure. Find $\int x^2 d\mu$ This is one of my homework problems:
Let $C$ be the Cantor set and $\mu$ be the uniform measure (with the usual Cantor distribution). Find $\int x^2 d\mu$.
I tried finding the value by using symmetry property, and failed, unsurprisingly. I... | Here is another solution. Let $X_1, X_2, \cdots$ be i.i.d. and $\mathsf{P}(X_i = 0) = \mathsf{P}(X_i = 2) = \frac{1}{2}$. Then
$$ Y = \sum_{m=1}^{\infty} \frac{X_m}{3^m} \tag{*}$$
has the distribution $\mu$. So it follows that
$$ \int x^2 \, d\mu
= \mathsf{E}[Y^2]
= \mathsf{Var}(Y) + \mathsf{E}[Y]^2
= \sum_{n=1}^{\inft... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2442274",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Probability of $2$ people being together out of a group of $4$. There were $2$ people who wanted to be in a group together. Those $2$ people were in a pool of $4$ people. Someone would randomly select $2$ people in a row and those people would be in a group together. The remaining $2$ would also be in a group.
What th... | We are placing the four people in two groups of two people. Notice that person $A$ must be in a group with one of the other three people, of whom only one is person $B$. Hence, the probability that persons $A$ and $B$ are placed in the same group is $1/3$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2442396",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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"answer_id": 1
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Is there a non-brute-force way to find $x$ such that $45\leq x < 200$, $x\equiv 0 \pmod{5}$ , $x\equiv1 \pmod{8}$, $x\equiv1 \pmod{12}$? Like the title says, I'm wondering if there's a non-brute-force way to determine $x$ such that
$$45\le x<200,$$
$$x\bmod5\equiv0,$$
$$x\bmod8\equiv1,$$
$$x\bmod12\equiv1$$
I know I ca... | Well:
*
*Since the number is $0$ mod $5$, we don't have to check all integers in $[45,200)$, but only the multiples of $5$: $45, 50, 55, \ldots$
*But we also know $x \mod{8} = 1$, which happens one in every 8 multiples of $5$. So $x \mod 40 = 25$. Now we only have to check $65, 105, 145, 185$.
*The third statement... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2442705",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Solve differential equation $y''-2y'+3y=\sin x$ using invers operator method.
$$y''-2y'+3y=\sin x$$
$s^2-2s+3=0$
$s=1 \pm i\sqrt {2}$
uisng operator $(D^2-2D+3)y_1=\sin x$
\begin{aligned}
y_1 &= \frac{1}{D^2-2D+3}\sin x \\
&= \Im\left(\frac{1}{D^2-2D+3} e^{ix}\right) \\
&= \Im \left(\frac{1}{i^2-2i+3}e^{ix}\right) ... | The starred part consists in basic algebraic manipulations of complex numbers, with $i^2 = -1$ and $e^{ix} = \cos x + i \sin x$. Indeed,
\begin{aligned}
\frac{1}{i^2 -2i + 3} & = \frac{1}{2 -2i}\\
& = \frac{1}{2} \frac{1}{1 - i} \frac{1+i}{1+i} \\
& = \frac{1}{2} \frac{1 + i}{2} \, .
\end{aligned}
Now, multiplying by $... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 0
} |
Prove that $\tau (x_1 x_2 ... x_k) \tau^{-1} = (\tau(x_1) \tau(x_2) ... \tau (x_k))$ I want to prove that $\forall \tau \in S_n$ and for pairwise different $x_1,...,x_k \in [n]$ it holds true that:
$\tau (x_1 x_2 ... x_k) \tau^{-1} = (\tau(x_1) \tau(x_2) ... \tau (x_k))$
I don't quite understand the notation. I assume ... | HINT:
What is $\tau(x_1 x_2 \ldots x_k)\tau^{-1}$ applied to $\tau(x_1)$ ?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2443045",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Proving $\int_{0}^\pi \frac{2\cos 2\theta + \cos 3\theta}{5+4\cos\theta} = \frac{\pi}{8}$ Given that $$\int_{|z|=1|}\frac{z^2}{2z+1} dz = \frac{i\pi}{4}$$,
show $$\int_{0}^\pi \frac{2\cos 2 \theta + \cos 3\theta}{5+4\cos\theta} = \frac{\pi}{8}$$.
I saw the bounds of the latter integral and thought that I should try a... | With $\theta\to-\theta$
$$
I=\int_0^{-\pi} \frac{2\cos(2\theta)+\cos(3\theta)}{5+4\cos(\theta)}\,(-)d\theta=\int_{-\pi}^0 \frac{2\cos(2\theta)+\cos(3\theta)}{5+4\cos(\theta)}\,d\theta
$$
\begin{align}
2I
&=\int_{-\pi}^\pi \frac{2\cos(2\theta)+\cos(3\theta)}{5+4\cos(\theta)}\, d\theta\\
&=\int_{|z|=1} \frac{{\bf Re\,}(2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2443161",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Prove that the open ball $B(0,1) \in \mathbb{C}$ is connected. Using the theorem, An open set $G \subset \mathbb{C}$ is connected iff for any two points $a,b$ in $G$ there is a polygon from $a$ to $b$ lying entirely in $G$.
I know an open ball is indeed an open set. I'm having trouble formulating the proof that if I ha... | Why don't you simply join $a$ and $b$ with a segment?
Let $a,b\in B(0,1)$ the $t\to ta+(1-t)b$ for $t\in [0,1]$ is a parametrization of the segment that joins $a$ and $b$. All the points along this segment are inside $B(0,1)$ because for all $t\in [0,1]$,
$$|ta+(1-t)b|\leq t|a|+(1-t)|b|< t\cdot 1+(1-t)\cdot 1=1.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2443252",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Show if this is valid or invalid (Propositional Logic) I just don't understand clearly what the question wants me to do?
It says:
Check if the following is valid or invalid
(I will take one question as an example.)
$$H\implies D $$
$$R\implies S$$
$$Therefore: (H \land R \implies S \land D)$$
I understand that they wa... | There are many 'laws of deduction': there are many different systems of deduction, each of which with their own set of laws or rules ... so it would be good to know which rules you are allowed to use.
Nevertheless, here is a proof using fairly commonly used rules:
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2443357",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Recurrence Relations and Linear-Feedback Shift Registers I have an exercise for a cryptography/number theory course that I'm trying to work on.
In the exercise, I have a Linear Feedback Shift Register which is working in mod 3 with digits {0,1,2}.
The LFSR is using a recurrence relation of degree 2, which looks lik... | Via method described in comments presuming each digit of $Z$ is a digit of $S$ $\mod 3$ I get the following;
$$C_0=2$$
$$C_1=1$$
$$S=22011022011$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2443509",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to prove that $e^{\gamma}={e^{H_{x-1}}\over x}\prod_{n=0}^{\infty}\cdots?$ How do we show that
$$e^{\gamma}={e^{H_{x-1}}\over x}\prod_{n=0}^{\infty}\left(\prod_{k=0}^{n}\left({k+x+1\over k+x}\right)^{(-1)^{k}{{n\choose k}}}\right)^{1\over n+2}?\tag1$$
Where $\gamma$ is the Euler-Mascheroni constant
$H_0=0$, $H_n$ ... | Here's a possible start
manipulating the double sum.
$$
\begin{align}\\
S(x)&= \sum_{n=0}^{\infty}{1\over n+2}\sum_{k=0}^{n}(-1)^k{n\choose k}\ln\left({k+x+1\over k+x}\right)\\
&=\sum_{n=0}^{\infty}{1\over n+2}
(\sum_{k=0}^{n}(-1)^k{n\choose k}\ln(k+x+1))-\sum_{k=0}^{n}(-1)^k{n\choose k}\ln(k+x)))\\
&=\sum_{n=0}^{\inft... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2443642",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Using set notation for sets with repeating characters Consider the following set:
{x, y, yy, yyyy, yyyyyyyy, yyyyyyyyyyyyyyyy, ...}
How would set builder notation be used to represent such a set? My understanding of the limitation is very limited as I only have experience using it for sets whose only elements are numbe... | First you want a notation for repeating a string n times.
Let s be a string and define by induction, s^1 = s and s^(n+1) = ss^n.
Your set appears to be { x, y, yy, y^(4n) : n in N }.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2443768",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Uniqueness of element Group element $a \in G$ have order 6 where $G$ can be any group. Prove that $a^3, a^4$ are the only elements $b,c$ in the group $G$ such that $a = bc$, $|b| = 2$, $|c| = 3$ and $bc = cb$.
Because $G$ can be any group, I'm really not sure how this could stand.
| Hint: Suppose that $b$ and $c$ are any such elements satisfying those assumptions. Squaring both sides of $a=bc$ yields $a^2=b^2c^2=c^2$. (This is where it is important that $b$ and $c$ commute.)
Use this to solve $c$ in terms of $a$, and that will then give you $b$ in terms of $a$. Mouse over below if you get stuck.
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2443911",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Bounding rapidly decreasing function by Schwartz function Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be continuous, and suppose that it is rapidly decreasing, i.e. for every $n\in\mathbb{N}$ we have
$$\sup\limits_{x\in\mathbb{R}}{|x^nf(x)|}<\infty.$$
Can we always find a Schwartz function $\varphi:\mathbb{R}\rightarrow\ma... | Sure with $g(x) = \sup_{y \in [x-1,x+1] } |f(y)|$ then $$g \ast \varphi(x) = \int_{-\infty}^\infty g(y) \varphi(x-y)dy$$ is Schwartz where $$\varphi(x) = 3 \, e^{-1/(1-x^2)} 1_{|x| < 1} \in C^\infty_c$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2444044",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Stationary point of a functional In the Euler-Lagrange equation we consider the functional $$ I[y,y',x] = \int_a^b F(y,y',x) \operatorname{dx}$$ where $y = y(x)$ is a function of $x$.
It textbooks the stationary point of the functional $I$ is given such that $$ \frac{\operatorname{d}}{\operatorname{d\alpha}}I[y+\alpha ... | In finding the function $y$ which makes $I$ stationary, we go about looking for which $y$ the first-order change in $I$ with respect to the function $y$ vanishes.
Hence a direct analogy can be made to stationary points of functions, as opposed to functionals.
The parameter $\alpha$ that we used is just a means by whi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2444177",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Bounds for $S_k = \sum_{i=1}^k {3k \choose 3i}$ Consider:
$$S_k = \sum_{i=0}^k {3k \choose 3i}.$$
Is it true that for all sufficiently large values of $k$, $S_k(1/2)^{3k} < 1/2$?
In general, for:
$$S_{c,k} {ck \choose ci} = \sum_{i=0}^k,$$
for integer $c > 1$.
Is it true that for all sufficiently large values of $k... | There actually is a closed form for this sum.
Let $\lambda \in \mathbb{C}$. By the binomial formula we know that
$$\sum_{i=0}^{3k} \binom{3k}{i} \cdot \lambda^i = (1+\lambda)^{3k}.$$
We can view this in terms of linear algebra: let
$$\begin{align*}
u & = \left[ \binom{3k}{0}, \binom{3k}{1}, \binom{3k}{2}, \ldots, \bino... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2444322",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Why is an injective relation called left-unique? Given a relation $R \subseteq A \times A$. Then, $R$ might be injective or left-unique. My question is a language question: why is an injective relation called "left-unique"? My feeling would rather to call them (wrongly!) "right-unique", since its "right" elements have ... | It makes sense, because an injective relation has the property that $aRb$ and $a'Rb$ forces $a = a'$. That is, you have a uniqueness property on the left side. This is not happening on the right side: uniqueness statements involve equating two things which a priori need not be equal, which is happening on the left; on ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2444430",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
How prove this binomial this problem from book,he say it is clear have$$\sum_{k=0}^n\sum_{j=0}^{n-k}\binom{n}{n-k}\binom{n-k}{j}(a+tk)^{n-j-1}(-1)^{n-k-j}(a+b)^j
=a^{-1}(a+b)^n$$ where $a.b$ be real numbers,
why it is clear? if not,how to prove it?
| $$
\begin{align}
&\sum_{k=0}^n\sum_{j=0}^{n-k}\binom{n}{n-k}\binom{n-k}{j}(a+tk)^{n-j-1}(-1)^{n-k-j}(a+b)^j\tag1\\
&=\sum_{k=0}^n\sum_{j=0}^n\binom{n}{n-k}\binom{n-k}{j}(a+tk)^{n-j-1}(-1)^{n-k-j}(a+b)^j\tag2\\
&=\sum_{j=0}^n\sum_{k=0}^n\binom{n}{n-j}\binom{n-j}{k}(a+tk)^{n-j-1}(-1)^{n-k-j}(a+b)^j\tag3\\
&=\underbrace{a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2444564",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Is it possible to drill a square hole using drill of special pattern? I have known one of the solutions which uses Reuleaux triangle as the drill, however the hole actually is a "square" with four round corner (brief info about this solution).
However, in another post, someone claims to improve the approach and gives a... | Square-Hole Drill in Three Dimensions
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2444673",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
To find the $\phi(x)$ for particular value of x.
Let $\phi(x)$ be solution of $$x=\int_{0}^{x}\exp(x-t)\phi(t)dt, \quad x\gt0.$$
Then $\phi(1)$ is given by
*
*-1
*0
*1
*2
Now I know how to verify whether a function is solution of integral equation or not but here I guess $\phi(x)=x$ gives $... | Use Laplace method and get $\dfrac{1}{s^2}=\dfrac{1}{s-1}{\cal L}(\phi)$. Then find $\phi$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2444832",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
$ \bigcap\limits_{n=1}^N I_n \neq \emptyset$ for all $ N \in \mathbb{N}$ implies that $ \bigcap\limits_{n=1}^\infty I_n \neq \emptyset $? How can I show that a sequence of closed bounded (Not necessarily nested) intervals $ I_1, I_2, I_3 ,\ldots$ with the property that $ \bigcap\limits_{n=1}^N I_n \neq \emptyset$ for ... | Let $I_{n}=[a_{n},b_{n}]$, where $a_{n}\leq b_{n}$. We go to prove
that $\sup_n a_{n}\leq\inf_n b_{n}$ by contradiction. Denote $a=\sup_{n}a_{n}$
and $b=\inf_n b_{n}$. Suppose the contrary that $a>b$. Choose $l\in(b,a)$.
Then there exists $n_{1}$ and $n_{2}$ such that $a_{n_{1}}>l$ and
$b_{n_{2}}<l$. Take $N=\max(n_{1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2445015",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 7,
"answer_id": 3
} |
Evaluate $\lim_{(x,y)\to(0,0)}\frac{\cos(xy)-1}{x^2y^2}$ Evaluate $\lim_\limits{(x,y)\to(0,0)}\dfrac{\cos(xy)-1}{x^2y^2}$
The limit does exist.
The only thing I can think of is let $t=xy$. Then our limit becomes
$$\lim_{t\to 0}\dfrac{\cos(t)-1}{t^2}=\lim_{t\to 0}\dfrac{\cos(t)-1}{t^2}\dfrac{(\cos(t)+1)}{(\cos(t)+1)}=\l... | You are basically done,
I would break the expression inside the limit into,
$$\frac{-1}{1+\cos t} \frac{\sin t}{t} \frac{\sin t}{t}$$
Then use the product rule for limits and the famous limit $\frac{\sin t}{t} \to 1$ as $t \to 0$.
Edit
As per the answer of @Steven Stadnicki, your limit is only equal to the single vari... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2445240",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 7,
"answer_id": 1
} |
What would be the "Size" of this space be for which we have the following bases We are talking about bases in class and I am wondering if you have the bases for
$ P_\infty $ such that the bases are $$ P_0 = 1, P_1=x, P_2=x^2, ... $$
essentially the x-terms in the Taylor series representation. How many functions would b... | The resultant space is similar to that of analytic functions, which are precisely those that can be written in the form
$$
f(x) = \sum_{i=0}^{\infty} a_n (x-x_0)^n
$$
However, around the point of expansion $x_0$, there is only a certain radius of convergence. Thus your space is one that contains many functions that do ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2445319",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Incomparable languages $B \nleq_T A$ and $A \nleq_T B$ I came across a problem to prove that there are sets $A$ and $B$ termed 'Turing-Incomparable' languages $B \nleq_T A$ and $A \nleq_T B$. The only languages I could think of are if $A$ and $B$ are disjoint regular expressions such as $B=1^*$ and $A=0^*$. This way, o... | Your example of symbol-disjoint languages doesn't seem relevant as we can still have a Turing computable biijection between the two.
In this context, the point of Turing reducibility, comparability, jumps and the likes is to acknowledge the effects of Turing degrees on set recognition by different oracle machines.
Your... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2445435",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Is the unit group of any finitely generated reduced $\Bbb Z$ algebra finitely generated? If $A$ is finitely generated commutative reduced $\Bbb Z$ algebra, must the unit group $A^{\times}$ be finitely generated?
The question is motivated by the Dirichlet unit theorem which says the unit group of the algebraic integer ... | "I think the case that $A$ is finite as $\mathbf Z$-module is always true".
Yes, it's true, it's even presented as "a generalization of the unit theorem" in §4.7 of P. Samuel's booklet on ANT. The particular case that $A$ is an integer domain is easy, because then, as you said, $A$ would be an order of some number fiel... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2445562",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 1,
"answer_id": 0
} |
the derivative of $n^x, n\in\mathbb{R}$ using the definition of derivative I know that the derivative of $n^x$ is $n^x\times\ln n$ so i tried to show that with the definition of derivative:$$f'\left(x\right)=\dfrac{df}{dx}\left[n^x\right]\text{ for }n\in\mathbb{R}\\{=\lim_{h\rightarrow0}\dfrac{f\left(x+h\right)-f\left(... | If you don't have a definition of the logarithm handy (or suitable properties taken for granted), you cannot obtain the stated result because the logarithm will not appear by magic from the computation.
Assume that the formula $n^x=e^{x \log n}$ is not allowed. Then to define the powers, you can work via rationals
$$n^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2445693",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
What Are All Separable Solutions of the Laplace Equation? Let us call a separable solution of the Laplace equation $\frac{\partial^2\Phi}{\partial x^2}(x,y)+\frac{\partial^2\Phi}{\partial y^2}(x,y)=0$ as a solution of the form $\Phi(x,y)=X(x)Y(y)$ which satisfies this equation. My question is that what are all such sol... | When you separate variables, you are only constructing some special solutions, you are not trying to get all solutions. That's why you do not worry too much when assuming that no function vanishes too often.
A posteriori, after you have constructed enough solutions with separate variables, you will see if you can obtai... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2445816",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Expected value in an urn problem In an urn there are $n$ red balls and $m$ blue balls. I extract them without replacement. Let $X$=time of first blue. What is $E(X)$?
I found PMF of $X$ and it is, if $k > n+1$
$$P(X=k) =\frac{n(n-1) \dots (n-k+1)m}{(m+n)(m+n-1) \dots (n+m-k+1)}$$
How could I evaluate $E(X)$?
Edit: I'm ... | Assume there are $r$ red and $b$ blue balls. We define the random variables as
$X_1$ : number of red balls preceding the first blue ball.
$X_i$ : number of red balls drawn following the appearance of the $(i-1)$th blue but before the appearance of the $i$th blue ball,$\quad i=2,3,...$
$X_{b+1}:$ number of red balls dra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2445903",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
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