Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
What is $x$, if $3^x+3^{-x}=1$? I came across a really brain-racking problem.
Determine $x$, such that $3^x+3^{-x}=1$.
This is how I tried solving it:
$$3^x+\frac{1}{3^x}=1$$
$$3^{2x}+1=3^x$$
$$3^{2x}-3^x=-1$$
Let $A=3^x$.
$$A^2-A+1=0$$
$$\frac{-1±\sqrt{1^2-4\cdot1\cdot1}}{2\cdot1}=0$$
$$\frac{-1±\sqrt{-3}}{2}=0$$
I... | Hints (why it's impossible in reals):
*
*$3^{x} \gt 0$ for any real $\forall x \in \mathbb{R}$
*$a + \cfrac{1}{a} \ge 2$ for any positive real $\forall a \in \mathbb{R}^+$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2128506",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 0
} |
Ball withdrawn from a box. White ball by kth draw. A box contains m white and n black balls. Balls are drawn at random one at a time without replacement. Find the probability of encountering a white ball by the kth draw.
This is our tutorial sheet's question and I don't know the answer. Thanks in advance.
Edit : I corr... | The problem is to find the probability of getting a white ball in $k$-th draw. May be for the first time or may be not... so we may have already gotten a white ball in first $k-1$ times. Therefore we have to consider them. If we consider '$i$' as no. of times a black ball can come in those $k-1$ times(as $k$-th is assu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2128603",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
If the length of a chord of a circle If the length of a chord of a circle with equation $x^2+y^2=100$ is $16$ units, how far is the chord from the centre?
My Attempt;
$$x^2+y^2=100$$
$$x^2+y^2=10^2$$
So, centre of circle $=(0,0)$.
How do I move further? Please help.
Thanks.
| Without loss of generality, suppose the chord is vertical, and to the right of the origin. Then its $x$ coordinate is the distance from the center. You also know that the endpoints of the chord, $(x,\pm8)$ are on the circle -- this gives you a quadratic equation in $x$ that you can solve:
$$ x^2+8^2=100 $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2128715",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove that the language {|M2 accepts , and M1 doesn't accept } isn't in RE I was asked to prove that the language {(M1),(M2)|M2 accepts (M1) , and M1 doesn't accept (M2)} isn't in RE, and not in co-RE.
I tried reductions but didn't seem one that works, I also tried variations of the proof that Ld isn't in RE, but didn... | We show reduction from $HP$ and $\overline{HP}$
$$ f(<M>,x) = <M_{\phi}>, <M_{x}>$$
$M_{x}$ on input $w$ will act as follows:
*
*run $M$ on $x$.
*accept.
$M_{\phi}$ just rejects every input word right away (so it's language is empty).
Now observe that if $M$ halts on $x$, then $L(M_{x})=\Sigma^{*}$ and otherwise... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2128859",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Principal ideal $(a) = R$ iff $a$ is a unit I was wondering if for a ring $R$ with $1$, the following is true:
The ideal $(a)$ generated by some element $a \in R$ is the whole ring ($(a) = R$) iff $a \in R^\times$.
The direction $a$ is unit $\Longrightarrow (a) = R$ is clear.
But what about the other direction? I suspe... | If $(a)=R$, then $1\in (a)$. In particular, $ra=1$ for some $r\in R$, and thus $a$ is a unit.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2128978",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How many 3-tuples satisfy $x_{1} + x_{2} + x_{3} = 11;$ $(x_{1} ,x_{2} ,x_{3}$ are nonnegative integers?) I know that the total number of choosing without constraint is
$\binom{3+11−1}{11}= \binom{13}{11}= \frac{13·12}{2} =78$
Then with x1 ≥ 1, x2 ≥ 2, and x3 ≥ 3.
the textbook has the following solution
$\binom{3+5−... | This can be solved also using the stars and bars method.
The point is paying attention to variables that take value 0.
So you have 3 cases:
1) all variables $\ne 0$
This amounts to $\binom{11-1}{3-1}=45$
2) just one variable have value $0$ (and hence two others are $\ne 0$)
This amounts to $\binom{3}{1}\cdot\binom{11-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2129086",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 3
} |
elementary set theory (being a member and subset of a set)? I have to prove or give a counterexample for these two statements:
For the following statements about sets $A$, $B$, and $C$, either prove the statement is true or give a counterexample to show that it is false.
A. If $A \in B$ and $B \subseteq C$, then $A \s... | This first statement is false.
A counter-example could be: $A=\{1\}$, $B=\{\{1\}\}$ and $C=\{\{1\},\{2\}\}$.
Then you have $A\in B$ and $B\subset C$, but you don't have $A\subset C$.
The second statement is true.
To prove it, take $A$, $B$ and $C$ meeting all the conditions. Then $A\in B\subset C$, so you do have $A\in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2129237",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
On an expansion of $(1+a+a^2+\cdots+a^n)^2$
Question: What is an easy or efficient way to see or prove that
$$
1+2a+3a^2+\cdots+na^{n-1}+(n+1)a^n+na^{n+1}+\cdots+3a^{2n-2}+2a^{2n-1}+a^{2n}\tag{1}
$$
is equal to
$$
(1+a+a^2+\cdots+a^n)^2\tag{2}
$$
Maybe this is a particular case of a more general, well-known re... | Hint:
Use synthetic division twice after you you've rewritten the expression as
$$\frac{(a^{n+1}-1)^2}{(a-1)^2}=\frac{a^{2n+2}-2a^{n+1}+1}{(a-1)^2}$$
$$\begin{array}{*{11}{r}}
&1&0&0&\dotsm&0&-2&0&0&\dots&0&0&1\\
&\downarrow&1&1&\dotsm&1&1&-1&-1&\dotsm&-1&-1&-1\\
\hline
\times1\quad&1&1&1&\dotsm&1&-1&-1&-1&\dotsm&-1&-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2129326",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 3
} |
Totient function: $φ(2^n)$ I was wondering how to go about finding $\varphi(2^n)$.
I know that $\varphi(2)=1$ and that $φ(mn) = φ(m)φ(n)$, but in this case having $\varphi(2^n) = \varphi(2\times2\times2\cdots\times 2)$ does not work since we end up with $1$ and this is not the answer.
| Take a prime $p$. By definition
$$\phi(p^n)=\#\{q\leq p^n,\,\,\gcd(p^n,q)=1\}$$
Let's count. Between $1$ and $p^n$ we have $p^{n-1}$ numbers who have $p$ in their prime factorisation and so
$$\phi(p^n)=p^n-p^{n-1}=p^{n-1}(p-1)$$
For $p=2$ we get $\phi(2^n)=2^{n-1}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2129420",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Using importance sampling to simulate the mean of a normal distribution truncated to interval [0,1] So in these notes it says that importance sampling is:
$$\int_F sf(s)ds = \int_G s \frac{f(s)}{g(s)}g(s)ds$$
And then it proceeds to give the following example:
In this example, if we draw from $f(x)$, are we effectiv... | This is a 'toy' example because it is easier and better to do
numerical integration, roughly as used to make printed normal tables,
to get the correct answer. However, it is a nice simple example to
get you acquainted with acceptance sampling.
Numerical integration (no simulation). So at the start, let's find the corre... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2129538",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Bayes's theorem from Tversky and Kahneman in Michael Lewis's The Undoing Project Michael Lewis's book "The Undoing Project" is concerned with the (mathematical) psychologists Daniel Kahneman and Amos Tversky. (Kahneman won the 2002 Nobel Prize; Tversky died in 1996.) On page 157, this question is quoted:
The mean IQ... | You expect the mean IQ of the remaining $49$ children to be $100$. Imagine that the first child gives one of his IQ points to each of the remaining $49$ children. Then we expect the remaining $49$ children have a mean IQ of $101$, and the first child now has an IQ of $101$, so all $50$ children together have a mean I... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2129646",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Help with Rank and Nullity of transpose matrices I'm stuck on this question;
Show that $Nul(A^tA) = Nul(A)$ for every matrix A.
I don't really know where to start on this. I know that rank is not changed by transposing, so nullity is also something I can figure out (rank nullity theorem).
But this is as far as I can ge... | You have to show that both sets are equal and equality of sets are shown through containment both ways.
It is clear that something in Null (A) is clearly in Null (A$^{t}$A).
Now the other way, if some vector x $\in$ Null (A$^{t}$A) then, A$^{t}$Ax = 0 which implies that x$^{t}$A$^{t}$A(x) = 0 which implies that,
(Ax... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2129781",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Show E$\big[\frac{\bar{X}(1-\bar{X})}{n}\big] = \frac{(n-1)p((1-p)}{n^2}$ I am trying to show E$\big[\frac{\bar{X}(1-\bar{X})}{n}\big] = \frac{(n-1)p((1-p)}{n^2}$, for $X_1, X_2,\ldots, X_n\sim\operatorname{b}(1,p)$.
I can get to E$\big[\frac{\bar{X}(1-\bar{X})}{n}\big]$ = $\frac{1}{n^2}(np) - (\frac{1}{n}E\big[\bar{X}... | If you know that $$n\bar X \sim \operatorname{Binomial}(n,p),$$ so that $$\operatorname{E}[n \bar X] = np, \quad \operatorname{Var}[n \bar X] = np(1-p),$$ then the calculation is straightforward: $$\begin{align*} \operatorname{E}\left[\frac{\bar X (1-\bar X)}{n}\right] &= \operatorname{E}\left[\frac{n \bar X}{n^2} - \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2129863",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Compute $\int_{-\infty}^{\infty} x^2e^{-x^2} \,dx$ $$\int_{-\infty}^{\infty} x^2e^{-x^2} \,dx$$
$g(x) = x^2e^{-x^2}$
Well, After computing it's fourier transform, which is $g(w) =\frac{2-w^2}{8\sqrt\pi}\cdot e^{\frac{-w^2}{4}}$.
In the solution they used some formula and said that:
$\int_{-\infty}^{\infty} x^2e^{-x^2} ... | Just for fun, here's a method that completely sidesteps integration by parts.
\begin{align}
\int_{-\infty}^\infty x^2e^{-x^2}dx&=-\int_{-\infty}^\infty\frac{\partial}{\partial\mu}e^{-\mu x^2}dx\bigg\vert_{\mu=1}\\
&=-\frac{d}{d\mu}\int_{-\infty}^\infty e^{-\mu x^2}dx\bigg\vert_{\mu=1}\\
&=-\sqrt{\pi}\frac{d}{d\mu}\left... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2129996",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
representation of delta function I want to prove that
$\delta(w) = \frac{1}{\pi^2} \int_{- \infty} ^{ \infty} \frac{dy}{y(y-w)}$
Could anyone help? I did the integration in two parts: $w=0$ and $w$ is not zero and I showed that for $w=0$, integral becomes infinite and for $w$ is not equal to zero it becomes zero. But... | It comes from the Hilbert transform but you have to be very careful about
how you define the integrals. You define:
$$ H(u)(t) = \frac{1}{\pi} {\rm P.V.} \int \frac{u(\tau)}{t-\tau} \; d\tau$$
A remarkable (and non-trivial) identity is that $H(H(u))(s)=-u(s)$ and your expression amounts to evaluating this for $s=0$. Th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2130155",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Radius of convergence of $\sum_{n=0}^{\infty} z^{n!}$? What is the radius of convergence of $\sum_{n=0}^{\infty} z^{n!}$?
I tried using ratio test and root test , applying the latter leaves me with the same type of problem again that is with $\sum_{n=0}^{\infty} z^{(n-1)!}$ ?.
| Hadamard's formula gives the answer at once:
If $\sum_n a_n zn$ is a power series, its radius of convergence $R$ is given by
$$\frac1R=\limsup_{n\to\infty} \bigl(\lvert a_n\rvert^{1/n}\bigr)=1.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2130255",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Determining a 4x4 homogeneous matrix for a 2D transformation? Given the vertices of 2 triangles, as written below, how do I find a 4x4 homogeneous transformation matrix to describe the transformation from the first triangle to the second?
$$Triangle 1 = T_1 = \{(0,0)(0,1)(1,0)\}$$
$$Triangle 2 = T_2 = \{(1,1)(2,2)(3,1... | The most likely reason to want a $4\times4$ matrix for this is because you want to leverage some technology which is geared towards 3d operations. So you can think of your 2d coordinates as embedded into a 3d space which in turn is represented using homogeneous coordinates.
In general an affine 2d operation would have ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2130466",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Determining that a certain prime integer is irreducible in the Gaussian integers Let $p\in \mathbb{Z}$ be prime such that $p \equiv 3 \bmod 4$. I've read in many places that $p$ must be irreducible in $\mathbb{Z}[i]$, but I can't see why.
Could someone please explain the reason?
| This is because of an elementary fact , that $x^2 \equiv -1$ modulo p has only solutions iff $ p \equiv 1 $ modulo 4. You can see this proof in Niven Zuckermann's book of Number Theory.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2130586",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Definition of subsequence I need help understanding the following definition:
We say that a sequence $b:\Bbb N\rightarrow S$ is a subsequence of a sequence $a:\Bbb N\rightarrow S$ if there exists a strictly increasing sequence $p:\Bbb N\rightarrow \Bbb N$ such that $b=a\circ p.$
So if I take for example that $a$ is ... | If $b$ is the even naturals, and $a$ is the naturals, then
$$
p:\mathbb{N}\rightarrow \mathbb{N}\\
k\mapsto2k
$$
and thus $a\circ p:\mathbb{N}\rightarrow\mathbb{N}=S$ is exactly the even terms in the sequence $a$. The increasing condition is to make sure what you get is really a "sequence" in the sense that you have s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2130688",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
} |
Continuity in a compact metric space.
Let $(X,d)$ be a compact metric space and let $f, g: X \rightarrow \mathbb{R}$ be continuous such that $$f(x) \neq g(x), \forall x\in X.$$
Show that there exists an $\epsilon$ such that $$|f(x) - g(x)| \geq \epsilon, \forall x \in X.$$
I'm assuming he means $\epsilon > 0$. Well... | Your flaw is in assuming that the values $f(x')$ and $g(x')$ are fixed; they are not, since $x'$ depends on $\varepsilon$.
To prove the claim you just need to observe that $|f-g|$ is a continuous function (since it is the composition of continuous functions) which is positive, and that it is defined in a compact set, s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2130775",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 1
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Geometry/Algebra: Model Building
My professor gave us a list of problems related to a project.
For the triangle below we have the bigger triangle have dimensions c for width and a + w + b. That would mean that a and b are related to x and y directly and the smallest triangle (middle) has dimensions c - (x+y) and w.
... | This question is beautiful.
Now some things should be obvious from this picture namely that your cost function is
$$C= C_L\Big[\sqrt{a^2+x^2}+\sqrt{b^2+y^2}\Big]+C_R\Big[\sqrt{w^2+(c-x-y)^2}\Big]$$
Which can be rewriten as
$$C_L\Big[a\sqrt{1+(\frac{x}{a})^2}+b\sqrt{1+(\frac{y}{b})^2}\Big]+C_R\Big[w\sqrt{1+(\frac{c-x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2131029",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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How to Disprove $A-(B-C)=A-(B\cup C)$? For $A,B,C$ sets. I know I want to show that it is not the case that $A-(B-C)$ is not equal to $A-(B\cup C)$, I also know that the definition of set deference is x is an element in A but not in B if I have set A-B, but to disprove this is getting a bit difficult.
| You can also disprove the statement by using truth tables, $A$ having the truth value $1$ (or TRUE) if $x \in A$ and $0$ (or FALSE) if not. The table for the union of sets $A$ and $B$, corresponds to the table of a logical OR (not exclusive), hence $A \vee B$ and set difference $A \setminus B$ corresponds to ($A$ AND ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2131128",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 2
} |
What are the group homomorphisms from $ \prod_ {n \in \mathbb {N}} \mathbb {Z} / \bigoplus_ {n \in \mathbb {N}} \mathbb {Z} $ to $ \mathbb {Z} $? By a theorem of Specker, there’s only the zero map since any map out of $ \prod_{n \in \mathbb{N}} \mathbb{Z} $ is determined by the values of the unit vectors, which all lie... | There is a nice quick proof. I'm not sure who the proof is due to.
The statement is equivalent to
If $P$ is the group of sequences ${\bf a}=(a_0,a_1,\dots)$ of integers, and $f:P\to\mathbb{Z}$ is a homomorphism that vanishes on finite sequences (so that $f({\bf a})=f({\bf b})$ whenever ${\bf a}$ and ${\bf b}$ differ i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2131320",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 1,
"answer_id": 0
} |
Is the function $\frac1x\log\sum\exp\left(c_i x^2\right)$ convex for every nonnegative $c_i$s? While reading a machine learning paper, I came across the following statement:
The function $\dfrac{f(x)}{x}$ is convex, where
$$f(x) = \log\left(\sum_{i = 1}^m \exp\left(c_i x^2\right)\right),$$
with $c_1, \dots, c_m \g... | Let me assume that $x \geq 0$ as otherwise the function is clearly not convex.
The perspective of the log-sum-exp function is convex, so
$$g(x,y) = \begin{cases} y \log\left(\sum\exp\left(c_i\frac{x}{y}\right)\right) & \text{if } y \geq \frac{1}{x}, x \geq 0 \\
\infty &\text{otherwise}\end{cases}$$
is convex too. The f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2131473",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Question about Cesàro summation Consider $$S_n = \sum_{i=0}^n a_i$$
and its Cesàro sums, defined as
$$ C = \lim_{n \to \infty} \frac1n\sum_{k=0}^n S_k$$
Is it always true that
$$ C = \lim_{n \to \infty} \frac1{L(n)}\sum_{k= n - L(n)}^n S_k$$
where $L(n)$ is any strictly increasing function such that $ 2 < L(n) < \ln(n)... | tl;dr: no.
Define for convenience $L(n)$; by assumption, we have $L(n)=o(n)$ as $n\to\infty$.
*
*Note that if $L(n)$ is bounded, then this is clearly false as the few missing constant terms do not really matter: you cannot hope that $\sum_{k=0}^n S_k$ be bounded in general (that is, there are Cesàro summable sums ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2131639",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Alternative Proof to irrationality of $\sqrt{2}$ using linear algebra I am taking my first Proof course, and have been researching alternative proofs to the irrationality of $\sqrt{2}$.
One that particularly interested me could be found on this site as number $10$, by D. Kalman.
To avoid butchering the format I'm not ... | The item linked to, proof number 10. It is a bit terse. I would add that the matrix $A$ is determinant $1.$ If there were any nonzero lattice points on the indicated line $L,$ there would be a lattice point $(c,d)$ with $0 < d < c,$ $c^2 = 2 d^2$ and minimal $c^2 + d^2 = 3 d^2 \neq 0.$ However, the part about the contr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2131774",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
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Why is the algebra of bounded (left) $F$-equivariant operators weakly closed in $B(\ell^2(F))$? Let $F$ be a free group with finite rank at least two. The Hilbert space of square-summable functions $f:F\to\mathbb{C}$ is denoted $\ell^2((F))$.
Define the weak (operator) topology on $B(\ell^2(F))$ as the topology induced... | Let $T$ be an $F$-invariant linear map, this is equivalent to
$$\langle gTx,y\rangle=\langle Tgx,y\rangle$$
for all $x,y\in \ell^2(F)$ and all $g\in F$. Note that the adjoint of the multiplication with $g$ is multiplication with $g^{-1}$:
$$\langle g\cdot x,y\rangle = \sum_{a\in F}\overline{x_{ga}} y_{a}=\sum_{a\in F}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2132125",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Vector subspace equality proof/disproof. Given $R,S,T$ are subspaces of vector space $V$, and $R+S=R+T$, does it follow $S=T$?
Please don't give a full proof, but some general help would be much appreciated. I get the basic idea that to show $S=T$ would be to show them to be subsets of one another. Not sure how to do t... | No. Think about the following example. V is the plane, R is a line which passes the origin, S, T are different lines which also pass the origin.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2132261",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
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numerically evaluate Schwarz Christoffel mapping The Schwarz Christoffel mapping is given by
$$f(\zeta) = \int_{\zeta_0}^\zeta \frac{1}{(w-z_1)^{1-(\alpha_1/\pi)}\cdots (w-z_n)^{1-(\alpha_n/\pi)} } \,\mathrm{d}w $$
where $z_i$ are complex points, and $\alpha_i \in (-\pi, \pi)$ are the corresponding angles.
To make plo... | Here is a way to use "integral" or "quad" to get numerical values with Matlab.
Let us take the example of function
$$\tag{1}f(z)=\int_{0}^z \dfrac{dw}{\sqrt{w^2-1}}$$
which is in fact identical to arcosh$(z)-i\pi/2$, or more exactly to $s$(arcosh$(z)-i\pi/2)$ where $s$ is the sign of $\Im(z)$, the imaginary part of $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2132485",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What is the decomposition of the ring $\mathbb{F}_p[x]/(x^n-1)$? Let $p$ be a prime, and $n\ge 1$ an integer.
I'd like to decompose the ring $\mathbb{F}_p[x]/(x^n-1)$ into a direct product of artinian local rings.
I know we can write $x^n-1 = \prod_{d\mid n}\Phi_d(x)$, but how do the cyclotomic polynomials $\Phi_d(x)$ ... | A natural way is to look at your ring as the ring of $n\times n$ circulant matrices over $\mathbb{F}_p$ (see e.g. Sect 2.3 here).
In this paper we have written down explicit decompositions for the group of units in such rings as an abelian group. I imagine this would help to deal with your question too.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2132619",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
Proving non-existence of rational points in a simple equation Working on Chapter 6.20 of Hammack Book of Proof
Show that $x^{2} + y^{2} - 3 = 0$ has no rational points.
First prove: IF $3\not\vert m$ THEN $m^{2}\equiv1(mod 3)$
If $3\not\vert m$ then either of two cases are true:
CASE $m\equiv 1 (mod3)$
$m=3k+1$ where... | Just do the hint.
If $n=2k $ then $n^2=4k^2\equiv 0\mod 4$
If $n=2k+1$ then $n^2=4k^2+4k+1\equiv 1 \mod 4$
So $3c^2 \equiv 0|3\mod 4$
And $a^2+b^2=0,1,2\mod 4$
So if $a^2 +b^2 =3c^2$ then all $a,b,c$ are even. But if we let $a=\gcd (a,b,c)a';b=\gcd (a,b,c)b';c=\gcd (a,b,c)c'$. $a',b',c'$ can't all be even (unless the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2132730",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Lower bound for $\frac{(x+y+z)^3}{xy+yz+zx}$ Let $x,y,z\geq 0$ and $x^2+y^2+z^2\geq 3$. What is the minimum value of $$D(x,y,z)=\frac{(x+y+z)^3}{xy+yz+zx}?$$
When $x=y=z=1$, $D(x,y,z)=9$. We have $(x+y+z)^2\leq 3(x^2+y^2+z^2)$ and $xy+yz+zx\leq x^2+y^2+z^2$, but these do not help directly with bounding $D(x,y,z)$.
| Let $x^2+y^2+z^2=k(xy+xz+yz)$.
Hence, by AM-GM we obtain:
$$\frac{(x+y+z)^3}{xy+xz+yz}\geq\frac{(x+y+z)^3}{xy+xz+yz}\sqrt{\frac{3}{x^2+y^2+z^2}}=$$
$$=\sqrt{\frac{3(x+y+z)^6}{(xy+xz+yz)^2(x^2+y^2+z^2)}}=\sqrt{\frac{3(k+2)^3}{k}}\geq\sqrt{\frac{3(3\sqrt[3]k)^3}{k}}=9.$$
The equality occurs for $x=y=z=1$, which says tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2132835",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to prove Vector Angles?
Any vector $\mathbf{v}$ is a unit vector if $\|\mathbf{v}\| = 1$.
Let $\mathbf{x}$, $\mathbf{y}$, and $\mathbf{z}$ be unit vectors, such
that $\mathbf{x} + \mathbf{y} + \mathbf{z} = \mathbf{0}$. Show that
the angle between any two of these vectors is $120^\circ$.
I know how to prove th... | We have that $$(\mathbf{x} + \mathbf {y} + \mathbf {z})^2 =0$$ $$\Rightarrow \mathbf {x}^2 + \mathbf {y}^2 + \mathbf {z}^2 +2 (\mathbf {x}\cdot \mathbf {y} + \mathbf {y}\cdot \mathbf {z} + \mathbf {z}\cdot \mathbf {x}) =0$$ $$\Rightarrow (\mathbf {x}\cdot \mathbf {y} + \mathbf {y}\cdot \mathbf {z} + \mathbf {z}\cdot \m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2132975",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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Find all real number satisfying $10^x+11^x+12^x = 13^x+14^x$
Find all real number $x$ satisfying $10^x+11^x+12^x = 13^x+14^x$
My Work
Dividing by $13^x$ we get
$$\left( \frac{10}{13} \right)^x +
\left( \frac{11}{13} \right)^x +
\left( \frac{12}{13} \right)^x
= 1 + \left( \frac{14}{13} \right)^x$$
The LHS is a ... | If you know that the solution must be an integer, this type of equation is known as exponential Diophantine, and there is no known formal procedure to solve it in the general case and probably none exists (this was proven for ordinary Diophantine equations).
If the solution is allowed to be real, there is no systematic... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2133084",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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What does the notation $\Gamma \vDash \phi$ mean (in Mathematical Logic)? What does the notation $\Gamma \vDash \phi$ mean (in Mathematical Logic)?
I understand that $\Gamma$ represents a system of formulas, and that $\phi$ represents an individual formula.
I also know that $\Gamma \vDash \phi$ means " $\phi$ is a sema... | Are you familiar with the notation $\mathfrak A\vDash \phi$, meaning that the structure $\mathfrak A$ satisfies $\phi$, or (in other words) $\phi$ is true in $\mathfrak A$?
$\Gamma\vDash \phi$ is then shorthand for:
$$ \forall \mathfrak A \bigl[ (\forall \psi\in\Gamma: \mathfrak A\vDash \psi) \;\to\; \mathfrak A\vDash\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2133202",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is my proof for this claim acceptable? (if f is continuous, than f is sequentially continuous) Let $S$ be a topological space, prove that if $f: S → \mathbb{R}$ is continuous at $a$ and a sequence $x_n∈ S → a$, then $f(x_n) → f(a)$.
We need to prove that, for any $ε > 0$, there exists an $N$ such that $|f(x_n)$ - $f(a)... | Your proof is logically correct, but working with $|x-a|<\delta$ you have tacitly assumed that $S$ is a metric space.
In the formulation of the problem $S$ is just a topological space. Hence you can talk about neighborhoods of points $a\in S$, open sets, etc., but there is no means to talk about a numerical size of suc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2133307",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Rational numbers - countability I have to show that the set of all finite sequences
$$
(q_1,q_2,\dotsc,q_k),\quad k \in \mathbb{N}
$$
of rational numbers is countable.
To prove that the set $\mathbb{Q}$ of all rational numbers is countable, I used that the set $\mathbb{Z}\times\mathbb{N}$ is countable and can be list... | You already have all the ingredients for your proof. You just need to assemble them.
Since you know $\mathbb{Z} \times \mathbb{N}$ is countable and that $\mathbb{N}$ and $\mathbb{Q}$ have the same cardinalty (countable) you know $\mathbb{Z} \times \mathbb{Q}$ is countable. Then you know that the subset
$\{0,1, \ldots, ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How do I find the point equidistant from three points $(x, y, z)$ and belonging to the plane $x-y+3z=0?$ I struggle to find the point ${P}$ equidistant from the points ${A(1,1,1), B(2,0,1), C(0,0,2)}$ and belonging to the plane ${x-y+3z=0}$.
Any tips?
| The line is perpendicular to the plane containing $A, B, C$. Transform the coordinates so the $ABC$ plane is the $x,y$-axis. Find the equidistant point on the $x,y$ plane. The line is the perpendicular to the plane at that point. Reverse the transform for that line.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2133525",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Maximizing summation function with upper bound as variable. I'm trying to find the general way of optimizing a summation function and I'm a bit lost. I would like to find the maxium (and the minimum, ideally) of:
$$ \sum_{n=1}^{x} f(n)$$
For example, let's say:
$$ f(n) = -(n-4)^2 + 16 $$
In this scenario, the answer is... | For a local max at positive integer $x$, you want $f(x) \ge 0$ while $f(x+1) \le 0$. Similarly for a local min with $\le$ and $\ge$. So you look for the places where $f$ changes sign.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2133655",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Showing matrices in $SU(2)$ are of form $\begin{pmatrix} a & -b^* \\ b & a^*\end{pmatrix}$ Matrices $A$ in the special unitary group $SU(2)$ have determinant $\operatorname{det}(A) = 1$ and satisfy $AA^\dagger = I$.
I want to show that $A$ is of the form $\begin{pmatrix} a & -b^* \\ b & a^*\end{pmatrix}$ with complex n... | Using @Omnomnomnom's suggestion $AA^\dagger =A^\dagger A$, we first obtain the relations
\begin{align}
AA^\dagger: r &= -\frac{su^*}{t^*}\ , \ u= -\frac{tr^*}{s^*} \\
A^\dagger A: r &= -\frac{tu^*}{s^*}\ , \ u= -\frac{sr^*}{t^*} \ .
\end{align}
Noticing the common factor $\frac{-t}{s^*}$ for $r_{A^\dagger A}$ and $u... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2133790",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Do any two coprime factors of $x^n-1$ over the $p$-adic integers $\mathbb{Z}_p$ which remain coprime over $\mathbb{F}_p$ generate comaximal ideals? Let $f,g$ be distinct irreducible factors of $x^n-1$ over $\mathbb{Z}_p[x]$ (polynomials over $p$-adic integers). Suppose $\overline{f},\overline{g}$ are coprime in $\mathb... | Suppose $(f,g)\ne 1$, then they are contained in some maximal ideal $m\supset (f,g)$, but the maximal ideals of $\mathbb{Z}_p[x]$ are precisely the ideals of the form $(p,h(x))$, where $h(x)$ is irreducible and remains irreducible mod $p$. Thus, $\mathbb{Z}_p[x]/m\cong \mathbb{F}_p[x]/(\overline{h})$. This implies that... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2133889",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Powerset functor on category Rel (sets and relations) Please will someone kindly explain how the powerset functor maps arrows in the category Rel. I understand that sets (objects) are mapped to their corresponding powerset, but I can't get my head around the arrows. If someone wouldn't mind giving a small example i'd b... | If $f:A\to B$ is a morphism in Rel, then it is relation between sets $A$ and $B, \ $ whose domain consists of certain elements $x$ of $A$, and whose codomain consists of certain elements $y$ of $B.$
Likewise, $\mathscr Pf:\mathscr PA\to \mathscr PB$ is a relation whose domain consists of certain elements $a$ of $\math... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2134001",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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Integrate $\int \frac{1}{(1-x)(1+x)}dx$ Integrate $$\int \frac{1}{(1-x)(1+x)}dx$$
$$\int \frac{1}{1-x^2}dx$$
$$=\tanh^{-1}(x)+C$$
When I look on Desmos though, this is only part of the answer?
The blue is the function that it is supposed to be, and the red is the derivative of the answer I got. As you can see it's rig... | $x=\pm 1$ are simple poles for the integrand function, in particular non-integrable singularities. That implies $\int_{a}^{b}\frac{dx}{1-x^2}$ has no meaning if $1$ or $-1$ belong to $[a,b]$. On the other hand,
$$ \frac{1}{1-x^2} = \frac{1}{2}\left(\frac{1}{1-x}+\frac{1}{1+x}\right)$$
clearly holds for any $x\neq \pm 1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2134095",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Differential of a Map I have the following map that embeds the Torus $T^2$ into $\mathbb{R}^3$:
$$f(\theta, \phi)=(cos\theta(R+rcos(\phi)),sin\theta(R+rcos(\phi)), rsin\phi)$$
noting that $0<r<R$.
I want to compute the differential of $f$, $f_*$, that maps $T_P(T^2)$ to $T_{f(p)}(\mathbb{R}^3)$.
This topic is extremely... | The differential of the map is given by the Jacobian. Basically what you want to do is take all of the partial derivatives of the coordinate functions and assemble them into a matrix. As you said this matrix should be a transformation from $T_p(T^2) \to T_{f(p)}(\mathbb{R}^3)$ so we want a $3 \times 2$ matrix. The matr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2134224",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Showing a set is convex I'm stuck on showing that the following set is convex.
$$\{x:\|x-x_0\|_2\leq \|x-y\|_2 \text{ for all } y\in S\},$$
where $S\subset \mathbb{R}^n$.
$\|x-x_0\|_2\leq \|x-y\|_2$ implies $(x_0-y)^Tx\geq 0$, which a half-space. Therefore, this set is equivalent to the intersection of half-spaces:
$$\... | Edit: Now that the question has been revised to ask about a different set, I'll give a different answer. Note that for a given vector $y$,
\begin{align}
& \| x - x_0 \|_2 \leq \| x - y \|_2 \\
\iff & \| x - x_0 \|_2^2 \leq \| x - y \|_2^2 \\
\iff & \| x \|_2^2 - 2 \langle x, x_0 \rangle + \| x_0 \|_2^2 \leq \| x \|_2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2134334",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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About multiplication operator on $L^p(X, \mu)$ where X is sigma finite Let X be sigma finite measure space and $\phi$ $\in$ $L^{\infty}(X, \Omega)$ and $M_\phi:L^p(X, \Omega)$ $\to$ $L^p(X, \Omega)$ multiplication operator then show that $\| M_\phi \|=\| \phi\|_{\infty}$.
My attempt:
I could prove that $\| M_\phi \| \l... | I think i have solved it. Can someone please check it..
$\| \phi \|_{\infty} = inf$ $\lbrace c>0 : \mu(\lbrace x\in X:|\phi(x)|>c\rbrace)=0\rbrace$
let $C_X=\big\lbrace c>0 : \mu(\lbrace x\in X:|\phi(x)|>c\rbrace)=0\big\rbrace$
now clearly $\| \phi \|_{\infty}- \epsilon \notin C_X$ So $\mu(\lbrace x\in X:|\phi(x)|>\| ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2134426",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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edges in a k-partite graph Let $G$ be a simple $k$-partite graph with parts of sizes $a_1$, $a_2$, ..., $a_k$. Show that $$m \le \frac{1}{2} \sum_{i=1}^{k}{a_i(n-a_i)}$$
How do I approach this problem? What is the relationship between edges and part sizes in a $k$-partite graph?
| Define $G = (V,E)$, with $V = \bigcup_{i=1}^k V_i$ and $V_r \cap V_s = \varnothing$ for $r \neq s$. Using the identity $$\sum_{v \in V}d(v) = 2m,$$ and the fact that $d(v) \leqslant n-a_i$ for $v \in V_i$,
we get
\begin{align}
m & = \frac{1}{2}\sum_{v \in V}d(v) \\
& = \frac{1}{2} \left(\sum_{v \in V_1}d(v) + \dots +\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2134644",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Solving 3 simultaneous equations It's been a while since I've had to do a simultaneous equation and I'm rusty on a few of the particulars.
Say for example I have the following equations:
x + y = 7
2x + y + 3z = 32
2y + z = 13
I know that I need to combine the above 3 equations into 2 other equations, for e... | Multiply equation (1) by 2 and subtract it from equation (2).
$2x + y + 3z - 2x - 2y = 32 - 14$
$-y +3z = 18$ ......(4)
Now solve equation (3) and (4) to find y and z.
Multiply equation (4) by 2 and add with equation (3).
$2y + z -2y + 6z = 13 - 36$
$7z = -23$
$z = \frac {-23}{7}$
Then from equation (1) find x.
Way to... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2134777",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Categories for which Natural Transformations are the Set of Arrows From Categories for the Working Mathematician pg. 43:
Theorem 1. The collection of all natural transformations is the set of arrows of two different categories under two different operations of composition, $\cdot$ and $\circ$, which satisfy the interc... | The horizontal category is:
*
*Objects are categories
*Morphisms are natural transformations
*The product of morphisms is horizontal composition $\circ$
*The identity for an object $C$ is $1_{1_{\mathcal{C}}}$
The vertical categtory is
*
*Objects are functors
*Morphisms are natural trasnformations
*The pro... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2134892",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Evaluate the integral $\int \frac{x^2(x-2)}{(x-1)^2}dx$
Find $$\int \frac{x^2(x-2)}{(x-1)^2} dx .$$
My attempt:
$$\int \frac{x^2(x-2)}{(x-1)^2}dx = \int \frac{x^3-2x^2}{x^2-2x+1}dx $$
By applying polynomial division, it follows that
$$\frac{x^3-2x^2}{x^2-2x+1} = x + \frac{-x}{x^2-2x+1}$$
Hence $$\int \frac{x^3-2x^2}{... | HINT: write your Integrand in the form $$x- \left( x-1 \right) ^{-1}- \left( x-1 \right) ^{-2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2135003",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 7,
"answer_id": 1
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Who will win the Matrix game? I have a matrix of size $N \times M$, There is a doll placed at $(1,1)$, which is the upper-left corner of the matrix. Two player makes an alternative turn.
If the doll is at $(x,y)$ then a player can move it to
$$ (x+1,y) \text{ or } (x-1,y) \text{ or } (x,y+1) \text{ or } (x,y-1) $$
onl... | If $N\times M$ is even, then cover the board (matrix) with $2\times 1$ dominoes. Then the first player will move from one square of the domino to the next one. The second player will have to move into a new domino. The first players wins (always has an available move)
If $N\times M$ is odd, then cover the board with $2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2135093",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Is it true, that $\mathbb{R} \times \mathbb{Q} \sim \mathbb{R}$? Just like in title, is it true that $\mathbb{R} \times \mathbb{Q} \sim \mathbb{R}$?
My answer would be yes, since $\mathbb{R}^2 \sim \mathbb{R}$.
| If by $\sim$ you mean "there is a bijection between", then the answer is yes, and your argument works fine through Cantor Bernstein's theorem : there is an obvious injection $\mathbb{Q}\to \mathbb{R}$ and so a sequence of injections as follows :
$\mathbb{R}\to \mathbb{R}\times \mathbb{Q} \to \mathbb{R}\times \mathbb{R}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What are hyperreal numbers? (Clarifying an already answered question) This question already has an answer here. That answer is abstract. Could you help me with some not-so-abstract examples of what the answerer is talking about? For example, give examples of hyperreal numbers which are written as numbers, if that is po... | Unfortunately, there is no "concrete" description of the hyperreals. For instance, there is no way to give a concrete description of any specific infinitesimal: the infinitesimals tend to be "indistinguishable" from each other. (It takes a bit of work to make this claim precise, but in general, distinct infinitesimals ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2135263",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
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Use mathematical induction to prove that the sum of the entries of the $k^{th}$ row of Pascal’s Triangle is $2^k$.
Use mathematical induction to prove that the sum of the entries of the $k$-th row of Pascal’s Triangle is $2^k$. Begin by proving that the row sum for any particular row is double that for the previous ro... | There problem go from $n$ to $n+1$:
$$S_{n+1}={n+1\choose 0}+{n+1\choose 1}+...+{n+1\choose n+1}$$
Using Stiefel's rule:
$${n+1\choose k}={n\choose k}+{n\choose k-1}$$
and note that
$${n+1\choose 0}={n\choose 0}, \quad {n+1\choose n+1}={n\choose n}$$
so,
$$S_{n+1}={n\choose 0}+\left[{n\choose 0}+{n\choose 1}\right]+\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2135585",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Complex integral of polynomial by conjugate of dz Good time of day everyone,
So I have this question: Let $C_r$ be the circle of radius $|z-a|=r$. Let P(z) be a polynomial. I need to show that the following integral along $C_r$ is $\int P(z) d\overline{z}=-2\pi r^2 P'(a)$.
I do not know how to start this, I do not unde... | Note that $(z-a)\left ( \bar{z}-\bar{a} \right )=r^2$. Differentiate this to get
$$d \bar{z} = -\frac{\bar{z}-\bar{a}}{z-a} dz = -\frac{r^2}{(z-a)^2} dz $$
Thus,
$$\oint_{|z-a|=r} d\bar{z} \, P(z) = -r^2 \oint_{|z-a|=r} dz \frac{P(z)}{(z-a)^2} $$
Now use Cauchy's integral theorem.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2135680",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Trouble understanding the definition of line segment Let $x,y\in \mathbb R$.
A common definition of Euclidean line segment $xy$ on $\mathbb R$ - real line - going from $x$ to $y$ is the set ${\{s:s=tx+(1-t)y:0 \leq t \leq 1}\}$.
Obviously, since we aren't considering directed line segments, but rather simply "line seg... | $ 0 \le t \le 1 \iff 0 \le 1-t \le 1$. And $t = 1-(1-t)$
So $xy = {\{s:s=tx+(1-t)y:0 \leq t \leq 1}\}$
$ = {\{s:s=(1-(1-t))x+(1-t)y:0 \leq 1-t \leq 1}\}$
$={\{s:s=uy+(1-u)x:0 \leq u \leq 1: u = 1-t}\}$
$={\{s:s=ty+(1-t)x:0 \leq t \leq 1}\}=yx$ .
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How many arrangements of the letters of RECURRENCERELATION have the vowels in alphabetical order? This is my attempt at it.
We first arrange the vowels in alphabetical order:
A E E E E I O U
The following wedges indicate where we can place consonants
^ A ^ E ^ E ^ E ^ E ^ I ^ O ^ U ^
There can be more than one letter ... | Bear with me.
Suppose there are $K$ ways to arrange the vowels. Then for any arrangement of letters there will be $K$ equivalent ways that have the exact same consonants in the exact same possitions but with the vowells in possibly different places. So for example there will be $K$ possible ways to make words of the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2135941",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Prove that a square matrix can be expressed as a product of a diagonal and a permutation matrix. I am having problems with this linear algebra proof:
Let $ A $ be a square matrix of order $ n $ that has exactly one nonzero entry
in each row and each column. Let $ D $ be the diagonal matrix whose $ i^{th} $
diagona... | Hint: For $(1)$, find a matrix $P(i,j)$ that swaps columns $i$ and $j$. Your permutation matrix will be a product of $P(i,j)$'s.
For $(2)$, try to convince yourself that when $D$ is diagonal, $D^{T}=D$. It's not too hard!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2136024",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What's the $\lim_{m\to\infty}\prod_{k=1}^m (1-e^{-kn})$? I try to show $\lim_{n\to\infty}\lim_{m\to\infty}\prod_{k=1}^m (1-e^{-kn})=1$. It seems we need to give a lower bound of $\lim_{m\to\infty}\prod_{k=1}^m (1-e^{-kn})$ depending on $n$ and as $n$ tends to infinity this lower bound tends to 1. I am trying to calcula... | $$\displaystyle \lim_{m\to\infty}\prod_{k=1}^m (1-e^{-kn}) \geq \lim_{m\to\infty}1 - \sum_{k = 1}^me^{-kn} = 1 - \frac{e^{-n}}{e^{-n} - 1} = \frac{e^n - 2}{e^n - 1}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2136153",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Integral non-injective substitution ($u=x^2+1$) Integrate:
$$\int x^3 \sqrt{x^2+1} dx$$
My solution:
Choose $u=x^2+1 \Leftrightarrow x = \pm \sqrt{u-1}$, then du = $2x dx$. Therefore: $$\frac{1}{2} \int (u-1)(u)^{1/2} du = \frac{1}{2} \int (u^{3/2} - u^{1/2})du$$
$$= \frac{1}{5} (x^2+1)^{5/2} - \frac{1}{3}(x^2+1)^{3/2... | Let's recall what a $u$ substitution is really doing in an indefinite integral. You are replacing $$\int f(x)dx $$ with $$ \int f(x(u))x'(u)du$$ and noting that if $\frac{d}{du}F(x(u)) = f(x(u))x'(u)$ then $F'(x) =f(x).$ In other words finding an anti-derivative for $f(x(u))x'(u)$ is the same as finding one for $f(x).$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2136264",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Why is $X^4-X^2+1$ reducible over $\Bbb F_5$? I have checked $X^4-X^2+1=0$ and got the solution that the polynom is never equal 0 so it should be irreducibel.
However it is.
My prof gave me that hint:
$X^4-X^2+1=(X^2+aX+b)(X^2+cX+d)$
Unfortunately that did not really help me. How can I solve that equation and where ... | When we say a polynomial $f(x)$ is irreducible, that means we cannot write it as $f(x) = g(x)h(x)$ unless one of $g$ or $h$ is constant. You are confusing this with meaning "the polynomial has no roots," which is different. Now if the degree of $f$ happens to be $1$, $2$, or $3$, these will be the same (because then at... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2136399",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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The set $\{\frac{\varphi(n)}n:n\in \Bbb N\}$ Let $f(n)=\varphi(n)/n$, where $\varphi$ is the totient function. Since $0<\varphi(n)\le n$ and
$$\lim_{n\to \infty}f(p_n)=1$$
(where $\{p_n\}$ is the increasing sequence of primes) and
$$\lim_{n\to\infty}f(n\#)=0$$
we know that $\limsup f(n)=1$ and $\liminf f(n)=0$. But is ... | Let $p_n$ denote the $n$-th smallest prime number. For each $\epsilon > 0$, choose $N$ such that $1/p_N < \epsilon$. Now consider $n_k = p_{N+1}\cdots p_{N+k}$ so that
$$ f(n_k) = \left(1 - \frac{1}{p_{N+1}}\right) \cdots \left(1 - \frac{1}{p_{N+k}}\right). $$
This gives
$$ |f(n_k) - f(n_{k-1})| \leq \frac{1}{p_{N+k}} ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the value of $\binom{2000}{2} + \binom{2000}{5} + \binom{2000}{8} + \cdots \binom{2000}{2000}$
Find the value of $\binom{2000}{2} + \binom{2000}{5} + \binom{2000}{8} + \cdots \binom{2000}{2000}$
I've seen many complex proofs. I am looking for an elementary proof. I know the fact that $\binom{2000}{0} + \binom{... | For $n\ge0$ let
$$a_n=\binom n0+\binom n3+\binom n6+\cdots=\sum_{k=0}^\infty\binom n{3k},$$
$$b_n=\binom n1+\binom n4+\binom n7+\cdots=\sum_{k=0}^\infty\binom n{3k+1},$$
$$c_n=\binom n2+\binom n5+\binom n8+\cdots=\sum_{k=0}^\infty\binom n{3k+2};$$
we seek the value of $c_{2000}.$ Observe that
$$a_n+b_n+c_n=2^n$$
and, f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2136738",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 2
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what is the smallest unit of a real number to which it is composed of? I searched for relevant questions but my point is different. For example take set of real numbers with usual order then is the immediate successor of one. Firstly, I ask that is there any such number?? If yes then how surprisingly we beleive existen... | There is always a real number between an two distinct real numbers $a$ and $b$. For example $\frac{a+b}{2}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2136807",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Subway & Graphs In the city there is a subway. You can get from any station to any other one. How can I prove that if we close one of the stations ( it can be picked),you won't be able to drive through it), we will be able to get from any station to any other one.
| We want to show that there is some stop that can be removed without disconnecting the graph. to do it, choose a stop, $s_0$ at random. Then, for a stop $s$, define $d(s)$ to be the length of the shortest path (in terms of the number of stops) from $s_0$ to $s$. Now let $s^*$ be a stop such that $d(s^*)$ is maximal. ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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algebraic closure of $\mathbb{Q}_p$ is not complete A paper I'm reading says that the algebraic closure of $\mathbb{Q}_p$ is not complete, by using for example the Baire theorem. Wikipedia says the Baire theorem says that a complete metric space is a Baire space (meaning a countable intersection of dense open sets rema... | The usual proof relies on Krasner's lemma. Let $ L/K $ be an infinite algebraic extension, where $ K $ is a perfect local field complete with respect to some nonarchimedean valuation. Then, there is $ a_1, a_2 , \ldots $ in $ L $ which are linearly independent over $ K $, and we may choose coefficients $ c_i $ for each... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Find the primitives of a given function Let $ f:(0,\infty )\rightarrow \mathbb{R}$ be a function such that:$$f(x)=\left ( \frac{1}{x^{2}}-\frac{1}{(x+1)^2} \right )\cdot \ln\left ( \frac{1}{x^{2}}+\frac{1}{(x+1)^2} +a\right ), \, a > 0$$
Find the primitives of $f$.
I've noticed that $\frac{1}{x^{2}}-\frac{1}{(x+1)^2} =... | The integral can be rewritten as a finite sum of elementary integrals :
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2137133",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Solving the congruence $x^2 \equiv 4 \mod 105$. Is there an alternative to using Chinese Remainder Theorem multiple times? I'm trying to solve $$x^2 \equiv 4 \mod 105.$$
This is of course equivalent to $$(x+2)(x-2) \equiv 0 \mod 105$$
which is also equivalent to the system of congruences
$$(x+2)(x-2) \equiv 0 \mod 3$$
... | There are a couple ways to optimize. First you need only compute half of the $8$ combinations since if $\,x\equiv (a,b,c)\bmod (7,5,3)\,$ then $\,-x\equiv (-a,-b,-c)\bmod (7,5,3).\ $ Then use CRT to solve it for general $\,a,b,c\,$ to get $\,x\equiv 15a+21b-35c\,$ Use that to compute those $4$ values. It's very easy
e.... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If $n$ is an even integer greater than $2$, then $2^n - 1$ is not a prime. Fairly new to Discrete Mathematics and I'm stumped on this one. So we're asked to prove:
If $n$ is an even integer greater than 2, then $2^n - 1$ is not a prime.
What I can come up with is that since $n > 2$, we know that $n$ is not prime sinc... | You are doing fine. Now that you have shown a factorization of $2^n-1$ the only thing that can go wrong is that one of the factors is $1$. So if $n \gt 2, \ldots$
A more general statement is that if $n$ is composite, $2^n-1$ is never prime. The reasoning is the same. If $n=ab$ with $a,b \gt 1$ then $2^n-1$ is div... | {
"language": "en",
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Can a torsion group and a nontorsion group be elementarily equivalent? I think that the direct product
$\prod_{n \in {\bf N}\setminus \{0\}} {\bf Z}/(n)$
and the direct sum
$\bigoplus_{n \in {\bf N} \setminus \{0\}} {\bf Z}/(n)$
are elementarily equivalent but am not sure how to prove it.
| Yes, a torsion group can indeed be elementarily equivalent to a non-torsion group.
We can get a (somewhat) explicit example using the following fact, together with Łoś's theorem:
If $G$ is torsion, but has elements of arbitrarily large finite order, then any nontrivial ultrapower over $\mathbb{N}$ of $G$ is non-torsi... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Average busy time with Poisson arrival We have a factory that can process jobs. Each job takes an hour to complete. Jobs arrive according to a Poisson arrival process, with a mean of $\lambda$ jobs per hour. If the factory is free when a job arrives, it accepts the job with probability $p$, independently of other jobs.... | I'm assuming that the factory can only process one job at a time, and that jobs arriving when the factory is busy don't get processed. If that isn't correct, please let me know.
You can use Poisson thinning to divide arriving jobs (if you want, a priori) into two separate streams: one containing "acceptable" jobs (a Po... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2137623",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Can trace product (of matrices) inequality be in SDP? We know the standard form SDP has linear equality constraints (p.168 of Convex Optimization, S. Boyd):
In my study, I derive a trace product inequality like
$$\langle B_i,X\rangle \leq d_i\quad i=1\ldots n$$
where
*
*$X\in \mathbb{R^{n\times n}}$
*$B_i\i... | You can put your inequality into the form of an equality by adding a slack variable and writing it as
$\langle B, X \rangle + s=d$
where
$s \geq 0$
To put this in matrix form, write your constraint as
$\langle A, Z \rangle = d$
where
$A=\left[
\begin{array}{cc}
B & 0 \\
0 & 1 \\
\end{array}
\right]$
and
$Z=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2137771",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Matrix norm of two hermitian matrices. Let A and B be two hermitian matrices.
Let $|||\cdot|||$ be any induced matrix norm.
I hope to find some upper bound inequalities or relationship of the matrix norm:
$|||iA - B|||$.
The only thing I know is that it is
$|||iA - B|||\leq |||A||| + |||B|||$.
Which of the following i... | None of the three inequalities you’ve shown hold in general for Hermitian matrices $A$ and $B$.
To demonstrate this, we can look at the case where $A$ and $B$ are $1\times 1$ real matrices (trivially Hermitian), in which case the induced matrix norm reduces to an absolute value. Writing $A=a$ and $B=b$ for $a,b\in \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2137902",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Logarithmic integration of two related terms How do I prove that
$$\int_{0}^{\infty} du \left(\frac{u^{2}}{(u+a)^{3}} - \frac{u^{2}}{(u+b)^{3}}\right) = \ln \left(\frac{b}{a}\right)?$$
| $\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newc... | {
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If $m$th term and $n$th term of arithmetic sequence are $1/n$ and $1/m$ then the sum of the first $mn$ terms of the sequence is $(mn+1)/2$
If $m$th term and $n$th term of arithmetic sequence are $1/n$ and $1/m$ respectively then prove that the sum of the first $mn$ terms of the sequence is $(mn+1)/2$.
My Attempt ;
$$... | I am assuming equations created by you as equation (1) and (2) respectively.
From equation (1),
$a = \frac 1n - (m - 1)d$
Put value of a in equation (2),
$\frac 1n - (m - 1)d + (n - 1)d = \frac 1m$
$\implies (-m + 1 + n - 1)d = \frac 1m - \frac 1n$
$$\implies (n - m)d = \frac {n - m}{mn}$$
$$\implies d = \frac 1{mn}$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2138109",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
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The Cauchy Principal value of a rational function with only real poles $\newcommand{\PV}{\operatorname{P.V.}}$I have a doubt about the Cauchy Principal Value of real rational functions.
$f: \mathbb{R}\rightarrow \mathbb{R}$ is a rational function with $\deg(\text{denominator})>\deg(\text{numerator})$.
$\{x_1,x_2,\ldots... | Daniel Fischer already notes in the comments that some condition must be
imposed on $f$ to assure that the principal value exists.
For example, $\int_{-\infty}^\infty dx/x^2 = \infty$, principal value or no:
the integrand is positive, so there's no cancellation when we remove
small symmetrical neighborhoods of the pole... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2138324",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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In right triangle $ABC$ ($\angle A=90$), $E$ is a point on $AC$.Find $AE$ given that...
In right triangle $ABC$ ($\angle A=90$), $AD$ is a height and $E$ is a point on $AC$ so that $BE=EC$ and $CD=CE=EB=1$. $\color {red} {Without}$ using trigonometric relations find $AE$.
I do have a solution USING trigonometric re... | Here is a proof with only Pythagoreans:
(as I see, Michael Rozenberg used also the altitude theorem of right triangles)
We have 5 eqns in 5 variables AE, AC, BA, BD, AD:
Obviously
$AE = AC - CE = AC - 1;$
and 4 Pythagoreans:
$$
AC^2 = - AB^2 + (BD+DC)^2 = - AB^2 + (BD+1)^2;\\
AC^2 = AD^2 + DC^2 = AD^2 + 1;\\
AD^2 + BD... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2138416",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the probability of using debit card Our professor gave this question as assignment, My question is does this question have all information needed to solve it? I asked her and she said it has all information but still I cannot figure out how I can solve it with just knowing the probability of using debit card in su... | The intended answer is mostly likely $72\%$. This is because to say that a consumer uses his/her debit cards "in only other stores or not at all" is logically equivalent to saying that he/she does not use his/her debit cards in supermarkets. So if $28\%$ of consumers do use their debit cards in supermarkets, then $72... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2138499",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$n$ students $k$ tutors find $F(n,k)$ so that there is no tutor sit next to other tutor I have the following question :
$n$ students $k$ tutors find $F(n,k)$ so that there is no tutor sit next to other tutor.
Assume the students are not distinguishable and also the tutors not distinguishable,
What I did
I looked at $i... | Presumably everyone is to be seated in a line. The $k$ tutors divide the students into $k+1$ possibly empty groups. Each of the internal groups must contain at least one student so that the two tutors on the boundary of the group are not seated next to each other. This accounts for $k-1$ of the students, so we need onl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2138632",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Closed form for $S= \sum\limits_{k=0}^n x^k \binom{n}{k}^2$ I am looking for a closed form for $\displaystyle S= \sum_{k=0}^n x^k \binom{n}{k}^2$. Does there exist such closed form?
| $$\sum_{k=0}^n x^k \binom{n}{k}^2=(x-1)^n \text{P}_n\left(\frac{x+1}{x-1} \right) $$
$\text{P}_n$ is a Legendre polynomial :
http://mathworld.wolfram.com/LegendrePolynomial.html
This is related to a form of series definition of Legendre polynomials :
$$\text{P}_n(z)=\left(\frac{z-1}{2}\right)^n \sum_{k=0}^n \left(\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2138745",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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$\int_{0}^{\frac{\pi}{4}}\frac{\tan^2 x}{1+x^2}\text{d}x$ on 2015 MIT Integration Bee So one of the question on the MIT Integration Bee has baffled me all day today $$\int_{0}^{\frac{\pi}{4}}\frac{\tan^2 x}{1+x^2}\text{d}x$$ I have tried a variety of things to do this, starting with
Integration By Parts Part 1
$$\frac{... | Six years too late but just for the fun of approximations
Knowing the series expansion of the integrate, it is simple to build $P_n$, the $[2n+2,2n]$ corresponding Padé approximant.
$$P_n=x^2\,\frac{1+\sum_{k=0}^n a_k\,x^{2k}}{ 1+\sum_{k=0}^n b_k\,x^{2k}}$$ which can in turn write
$$P_n=\frac{a_n}{b_n} x^2\frac{\prod_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2138866",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "64",
"answer_count": 7,
"answer_id": 3
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Finding the original equation with Linear Law Two variables $x$ and $y$ are related by a certain equation. This equation may be expressed in two forms suitable for drawing straight line graphs. The two graphs shown have different variables plotted at each axis. Given the coordinates of a point on each line, find the or... | From the second graph we have,
$$x=m\frac{y}{x}+b$$
This is useful be we can solve for $y$. To get,
$$\frac{x^2-bx}{m}=y$$
So $y$ is a quadratic in $x$, we also have from the second graph that one point on our quadratic is given by,
$$x=6$$
$$\frac{y}{x}=2 \implies y=6(2)=12$$
So one point on our quadratic is $(6,12)$.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2138951",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove, by contradiction, that, if $cx^2 + bx + a$ has no rational root, then $ax^2 + bx + c$ has no rational root. Proposition: Suppose that $a$, $b$, and $c$ are real numbers with $c \not = 0$. Prove, by contradiction, that, if $cx^2 + bx + a$ has no rational root, then $ax^2 + bx + c$ has no rational root.
Hypothesis... | Since you have proved that $ap^2+bpq \neq 0$ and $cq^2 \neq 0$, you are very close to the answer.
Notice that $p(ap+bq)=-cq^2$. But $p(ap+bq) \neq 0 \Rightarrow p \neq 0$.
Divide both sides by $p^2$. Then you get $a+b\frac qp=-c\frac {q^2}{p^2}.$
This shows that $cx^2+bx+a=0$ has a rational root $\frac qp$. WHICH is co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2139056",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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How to solve this convex optimization problem (with absolute and linear objective function)? I have the following problem:
\begin{align*}
\sup_y&\quad \big | \langle u,y \rangle\big |\\
\mbox{s.t.}&\quad \frac{1}{2}\langle y,y \rangle\ + \langle b,y \rangle\ \geq \gamma.
\end{align*}
where $u$ is a constant vector.
I... | Given $\mathrm c, \mathrm y \in \mathbb R^n$ and $\rho > 0$,
$$\begin{array}{ll} \text{supremize} & | \mathrm c^{\top} \mathrm x |\\ \text{subject to} & \| \mathrm x - \mathrm y \|_2 \geq \rho\end{array}$$
Since $-\| \mathrm c \|_2 \| \mathrm x \|_2 \leq \mathrm c^{\top} \mathrm x \leq \| \mathrm c \|_2 \| \mathrm x \|... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2139110",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Solve an integral $\int\frac{\cos^3 x}{\sin^3 x+\cos^3 x}dx$ Solve an integral $$\int\frac{\cos^3 x}{\sin^3 x+\cos^3 x}dx$$
I tried to divide the numerator and denominator by $\cos^4 x$ to get $\sec x$ function but the term ${\sin^3 x}/{\cos^4 x}$ gives $\tan^2 x\sec^2 x\sin x$. How to get rid of $\sin x$ term?
| Let $$I = \int \frac{\cos^3x}{\sin^3x+\cos^3x} dx$$ $$I_1 = \int\frac{\sin^3x+\cos^3x}{\sin^3x+cos^3x}dx = x + C$$ and $$I_2 = \int\frac{\cos^3x-\sin^3x}{\sin^3x+\cos^3x}dx$$Then $$I = \frac{I_1 + I_2}{2}$$
$$I_2 = \int\frac{(\cos x-\sin x)(1+\frac{\sin2x}{2})}{(\sin x+\cos x)(1-\frac{\sin 2x}{2})}dx$$
Now substitute $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2139198",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Find all non-negative integers satisfying the conditions
Question 1. Find all non-negative integer a, b, c,d, e such that $$ a+b+c+d+e = 8$$
Question 2. Find all non-negative integer a, b, c,d such that $$ a+b+c+d = 8$$
Question 3. Find all non-negative integer a, b, c such that $$a+b+c = 8$$
Is there any meth... | The problem is asking to find all the partitions (with a specified number of parts).
You want to partition the number 8 in 5, 4, and 3 pieces, so the answer is in the section "Restricted part size or number of parts" of that wikipedia page.
Here is a maple example
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2139283",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
} |
A series involve combination I want find another Idea to find sum of $\left(\begin{array}{c}n+3\\ 3\end{array}\right)$ from $n=1 ,to,n=47$
or $$\sum_{n=1}^{47}\left(\begin{array}{c}n+3\\ 3\end{array}\right)=?$$ I do it first by turn $\left(\begin{array}{c}n+3\\ 3\end{array}\right)$ to $\dfrac{(n+3)(n+2)(n+1)}{3!}=\dfr... | By the well known hockey stick identity
$$ \sum_{n=0}^{47}\binom{n+3}{3} = \binom{47+3+1}{3+1} $$
and the problem is trivial from there.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2139408",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Finding the eigenvectors of a reflection matrix How could I find the eigenvectors of the following matrix?
$$\left(\begin{matrix} \cos(\theta) & \sin(\theta)\\ \sin(\theta)& -\cos(\theta)\end{matrix}\right)$$
I found the eigenvalues $1$ and $-1$ but I'm struggling with calculating the eigenvectors.
| Let $$A = \left(\begin{matrix}\cos(\theta) & \sin(\theta)\\ \sin(\theta)& -\cos(\theta)\end{matrix}\right).$$
$\lambda = +1,-1$ as you have already mentioned.
So eigenvectors corresponding to $\lambda = 1$ is got by solving for $x$ in $$Ax = x.$$
So do the row echelon conversion (or any other method to solve the simult... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2139518",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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} |
Defining $R\times R$ as a ring? I feel a bit stupid, but I know that the normal definition of $R\times R$ as $R \times S = \{(r, s) : r \in R, s \in S\}$, under $(r, s) + (r', s')=(r+r',s+s')$ and $(r, s) \cdot (r', s')=(rr', ss')$ is a ring.
But, can you define $R \times R$ otherwise as a ring?
I'm trying to decide ... | Hint: You can define multiplication in $R \times R$ by thinking about the complex numbers.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2139605",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
} |
Alternate method for reducing first order ODEs to become separable So I'm struggling to come to grips with a method that my prof introduced in a lecture. It's basically reducing ODEs so that they're separable and easier to work with.
Apparently the trick in this case is if you have an ODE that doesn't appear separable ... | $$y'=(4y-x-6)^2$$
$$z(x)=4y-x-6 \implies y'=z^2$$
$$z'=4y'-1 \implies z'=4z^2-1$$
$$\int \frac{dz}{4z^2-1}=\int dx$$
$$x=\frac{1}{4}\ln\left|\frac{1-2z}{1+2z} \right|+c$$
$$z=\frac{1}{2}\:\:\frac{1-e^{4(x-c)}}{1+e^{4(x-c)}}$$
$$y=\frac{1}{8}\:\:\frac{1-e^{4(x-c)}}{1+e^{4(x-c)}}+\frac{x+6}{4}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2139667",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How is a Generator Matrix for a (7, 4) Hamming code created? I see that a generator matrix is created with the following formulae:
$$G = \left[I_{k}|P\right]$$
I do not understand what P is in this case. In my notes, I am told that in a (7, 4) Hamming code situation my
$$G =
\begin{pmatrix} 1 & 0 & 0 & 0 & 1 & 0 & 1 \\... | The standard way of finding out the parity matrix $G_{k,n}$ for a Hamming code is constructing first the check parity matrix $H_{n-k,n}$ in systematic form.
For this, we recall that a Hamming code has $d=3$ (minimum distance). Hence the columns of $H$ have the property that we can find a set of $3$ linearly dependent c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2139807",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 5,
"answer_id": 0
} |
Arrange $m$ people in $m+r$ seats around a round table. The answer is, according to the book: $(m-1)!\cdot \binom{m+r-1}{r}$
I get why this is true. You arrange $m$ people in their seats in $(m-1)!$ ways and then you put $r$ empty spots inbetween them.I was thinking - choose $m$ seats out of $m+r$ to put the people on ... | With your way, you over count. If one group of $m$ seats that you choose, on the table, is a rotated version of another group of $m$, then you can get the same order, by just rotating them.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2139932",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Is $\nabla^{n}$ a correct terminology for a partial derivative? I'm learning optimization techniques and came accross the gradient (nabla) operator : $\nabla$.
If I'm right, the $\nabla$ operator of a function means the vector of all its partial derivatives. Then, if for example I'm talking about this specific partial ... | I don't know that anyone calls this the nabla operator. The symbol is nabla, but the operator is called the gradient. There is no need for a special notation for an individual partial derivative, you can just write $\partial f/\partial y$ or $\partial/\partial y$ for the operator.
You do sometimes see $\nabla_x f$ fo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2140189",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Existence of smooth coordinate charts such that composition map is smooth The following problem is problem 2.1 from John Lee's Introduction to Smooth Manifolds:
Define $ f: R \to R $ by $ f(x) = 1 $ if $ x \geq 0 $ and $ f(x) = 0 $ if $ x < 0 $. Show that for every $ x \in R $, there are smooth coordinate charts $ (U, ... | Only $x=0$ poses a problem. Here is what you can do at $x=0$:
Take $U=\mathbb R, \phi=Id:\mathbb R\to \mathbb R$ and $V=(0,\infty), \psi=Id:(0,\infty) \to (0,\infty)$.
Then $U \cap f^{-1}(V)=[0,\infty)$ and $ \varphi(U \cap f^{-1}(V)) $ to $ \psi(V) $ is the map $$[0,\infty) \to (0,\infty): r\mapsto 1$$ which is s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2140314",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to calculate 15(7)? I want to consider the value of the 'function' $15$ at the point $(7)$ of $\text{Spec}(\Bbb Z)$.
So we consider $15\in\Bbb Z$ over the composition:
$$\Bbb Z\to \Bbb Z/(7)\to k(7)$$
Where the last term is the residue field at $7$.
So we have $15\mapsto [1]\in\Bbb Z/(7)$, and I take it that $k(7)=... | Let's back up a bit. In general, if $A$ is a commutative ring and $P\subset A$ is a prime ideal, the residue field $k(P)=A_P/PA_P$ has a map $f:A/P\to k(P)$ defined by $f(a+P)=\frac{a}{1}+PA_P$. In other words, we take the canonical inclusion map $A\to A_P$ (taking $a\in A$ to the fraction $\frac{a}{1}$), and compose... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2140459",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Counterintuitive examples in probability I want to teach a short course in probability and I am looking for some counter-intuitive examples in probability. I am mainly interested in the problems whose results seem to be obviously false while they are not.
I already found some things. For example these two videos:
*
*... | The Shooting Room Paradox
A single person enters a room and two dice are rolled. If the result
is double sixes, he is shot. Otherwise he leaves the room and nine new
players enter. Again the dice are rolled, and if the result is double
sixes, all nine are shot. If not, they leave and 90 new players enter,
and so on (t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2140493",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "198",
"answer_count": 31,
"answer_id": 24
} |
Infinite sum of random variables
Let $\xi_i$ be independent discrete random variables such that $P[\xi_i = k] = {1 \over 10}$ for all $i \in \mathbb{N}$ and $k = 0,\dots,9$. Define
$$
X = \sum_{i = 1}^\infty \xi_i{1 \over 10^i}.
$$
What's the distribution of $X$?
The sum converges as every summand is less than ${1... | $\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2140610",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 4
} |
How can $\int \frac{dx}{(x+a)^2(x+b)^2}$ be found? Could you please suggest any hints or methods for solving $\int \frac{dx}{(x+a)^2(x+b)^2}$. I have used partial fractions to solve this integral but it is too long and complex solution. I'd like to know a simpler solution.
EDIT: $a\not= b$
| HINT:
Use $a-b=x+a-(x+b)$ and $$(a-b)^2=\{(x+a)-(x+b)\}^2=\cdots$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2140709",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Final step (Frullani's formula) The integral is:
$$I =\int_{0}^{\infty} \frac{\sin(\alpha x)\cos(\beta x)\cos(\gamma x)}{x}dx $$
My solution is:
$$ I=\frac{1}{2}\int_{0}^{\infty}\frac{\sin((\alpha-\beta)x)\cos(\gamma x)}{x}dx + \frac{1}{2}\int_{0}^{\infty}\frac{\sin((\alpha+\beta)x)\cos(\gamma x)}{x}dx$$
By applicati... | I'm pretty sure I saw this question the other day, but anyways. For a lot of parameters $\alpha$, $\beta$ and $\gamma$ the result is indeed $0$. But, for example, for $\alpha=\beta=\gamma=1$, you get
$$
\int_0^{+\infty}\frac{\sin x\cos^2x}{x}\,dx=\frac{\pi}{4}.
$$
So, a tip: Write
$$
\sin\alpha x\cos\beta x\cos\gamma x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2140835",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Minimize $\big(3+2a^2\big)\big(3+2b^2\big)\big(3+2c^2\big)$ if $a+b+c=3$ and $(a,b,c) > 0$
Minimize $\big(3+2a^2\big)\big(3+2b^2\big)\big(3+2c^2\big)$ if $a+b+c=3$ and $(a,b,c) > 0$.
I expanded the brackets and applied AM-GM on all of the eight terms to get :
$$\big(3+2a^2\big)\big(3+2b^2\big)\big(3+2c^2\big) \geq 3\... | More way.
Let $a+b+c=3u$, $ab+ac+bc=3v^2$ and $abc=w^3$.
Hence, we need to prove that
$$(3u^2+2a^2)(3+2b^2)(3+2c^2)\geq125u^6$$ or
$$27u^6+18u^4(9u^2-6v^2)+12u^2(9v^4-6uw^3)+8w^6\geq125u^6$$ or $f(w^3)\geq0$, where
$$f(w^3)=2w^6-18u^3w^3+16u^6-27u^4v^2+27u^2v^4.$$
But $f'(w^3)=2w^3-18u^3<0$, which says that $f$ is a de... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2141009",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 3
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