Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
find extreme values of $\frac{2x}{x²+4}$ I am doing my homework like a good little boy. I know that when I want to find the extreme values of a function I have to put the derivative equal to zero so I can find the x values. I've done it before with easier functions where I only have to use the power rule.
But with the ... | Here is an algebraic way without using calculus
$$\text{Let }y=\frac{2x}{x^2+4}\iff x^2y-2x+4y=0$$ which is a Quadratic equation in $x$
As $x$ is real, the discriminant of the above equation must be $\ge0$
i.e, $(-2)^2\ge 4\cdot y\cdot 4y\iff y^2\le \frac14$
We know, $x^2\le a^2\iff -a\le x\le a$
Alternatively, le... | {
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math fallacy problem: $-1= (-1)^3 = (-1)^{6/2} = \sqrt{(-1)^6}= 1$? I know there is something wrong with this but I don't know where. It's some kind of a math fallacy and it is driving me crazy. Here it is:
$$-1= (-1)^3 = (-1)^{6/2} = \sqrt{(-1)^6}= 1?$$
| The problem here is that the square root function, $\sqrt{-},(-)^\frac{1}{2}$, is not a single-valued function.
As PVAL says, it is a two-valued function, meaning you have two consistently choose which square root you're talking about. That's why you often will have problems when you have chains of equalities as above.... | {
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Separating 18 people into 5 teams A teacher wants to divide her class of 18 students into 5 teams to work on projects,
with two teams of 3 students each and three teams of 4 students each.
a) In how many ways can she do this, if the teams are not numbered?
b) What is the probability that two of the students, Mia and M... | The answer to the first question is just a multinomial coefficient divided by $2! \cdot 3!$ to make up for the fact that the teams are not numbered and we have therefore counted each partition into teams multiple times:
$${\binom{18}{3, 3 ,4,4,4}}/(2!\cdot 3!) = \frac{18!}{(3!)^2 * (4!)^3 \cdot (2! \cdot 3!)}=1.072. ... | {
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Interpreting a singular value in a specific problem In a similar spirit to this post, I pose the following:
Contextual Problem
A PhD student in Applied Mathematics is defending his dissertation and needs to make 10 gallon keg consisting of vodka and beer to placate his thesis committee. Suppose that all committee memb... | I think it is easiest to interpret this when we don't think of the vector spaces in question as $\mathbb{R}^2$, but rather, think of $A$ as a linear map between the vector space $V = (\text{beer}, \text{vodka})$ to the vector space $W = (\text{volume}, \text{alcohol content})$. This map takes the drinks you have, and ... | {
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"timestamp": "2023-03-29T00:00:00",
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Subgraphs of Complete graphs I have been studying a little graph theory on my own and a simple google search has not helped so I am deciding to turn to math stack exchange.
My question is: Given a complete graph $K_{n}$ where $n\ge 2$, how many non-isomorphic subgraphs occupy it?
My initial thought was that the formula... | The number of subgraphs (including the isomorphic subgraphs and the disconected subgraphs) of a comple graph (with n>=3) is
$$ \sum_{k=1}^n {n \choose k} ( 2^{k \choose 2} ) $$
I found it in Grimaldi, R. P. (2003) Discrete and Combinatorial Mathematics. (5th ed.) Pearson. Chap. 11. Solutions book.
I'm... | {
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Why is the supremum of the empty set $-\infty$ and the infimum $\infty$? I read in a paper on set theory that the supremum and the infimum of the empty set are defined as $\sup(\{\})=-\infty$ and $\inf(\{\})=\infty$. But intuitively I can't figure out why that is the case. Is there a reason why the supremum and infimum... | In addition to the reasons mentioned above about it making things easier, it also makes sense intuitively if you think about how each is defined. The supremum is the lowest upper bound on a set, so, since any real number is an upper bound on the empty set, no real number can be the lowest such bound (If x is that bound... | {
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Prove that: $ \cot7\frac12 ^\circ = \sqrt2 + \sqrt3 + \sqrt4 + \sqrt6$ How to prove the following trignometric identity?
$$ \cot7\frac12 ^\circ = \sqrt2 + \sqrt3 + \sqrt4 + \sqrt6$$
Using half angle formulas, I am getting a number for $\cot7\frac12 ^\circ $, but I don't know how to show it to equal the number $\sqrt2 +... | $$\text{As } \cot x =\frac{\cos x}{\sin x}$$
$$ =\frac{2\cos^2x}{2\sin x\cos x}(\text{ multiplying the numerator & the denominator by }2\cos7\frac12 ^\circ)$$
$$=\frac{1+\cos2x}{\sin2x}(\text{using }\sin2A=2\sin A\cos A,\cos2A=2\cos^2A-1$$
$$ \cot7\frac12 ^\circ =\frac{1+\cos15^\circ}{\sin15^\circ}$$
$\cos15^\circ=\cos... | {
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In how many ways can $r$ elements be chosen from $n$ elements with repetition, such that every element is chosen at least once? Thank you for your comments -- hopefully a clarifying example --
In choosing all combinations of 4 elements from (A,B,C), (e.g., AAAA,BBBB,ACAB,BBBC, etc.) how many of these combinations inclu... | Thw answer in general is $r!S(n,r)$, where $S(n,r)$ is the Stirling number of the second kind.
| {
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How to decompose permutations? In algebra, we have seen theorems such as
Every permutation is the product of disjoint cycles of length $\geq 2$.
I don't really know how to apply this, so I looked at its proof hoping it would be helpful.
Proof: Decompose $\{1, \dots, n\}$ disjointly in orbits of $\langle \sigma \ran... | The key to decomposing cycles is to trace the "orbit" of each element under the permutation.
So, for example, let's decompose
$$
\sigma= \begin{pmatrix} 1 & 2 \end{pmatrix} \begin{pmatrix} 3 & 4 \end{pmatrix} \begin{pmatrix} 1 & 2 & 3 \end{pmatrix}
$$
We begin by finding $\sigma(1)$. Applying the permutations from ri... | {
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Definition of limit and axiom of choice In the definition of limit of a function ($\epsilon-\delta$ definition) we say certain statements such as for every $\epsilon>0$ there exist $\delta>0$ ....
Now my question is, is a choice function required to ensure that for every $\epsilon$ there exist a $\delta>0$? Moreover, h... | The question doesn't make much sense, and it seems that you are misunderstanding the definition.
We define continuity by $\varepsilon$-$\delta$ at a point $x$ in such way. If you want to show that a function is continuous at $x$ it suffices to show that every $\varepsilon$ has such $\delta$. We are not required to assi... | {
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How can I calculate the points of two lines that connect two circles? Let's say I have two circles of equal or differing radii, a variable distance apart.
I want to calculate the end points of two lines, that will "connect" the circles.
And no matter how the circles may be oriented, they should still "connect" in ... | In either way if you know the circles find gradient of circle meaning y' and equate them so you get the line.
"this line will be tangent since they are not cutting circle"
Note: To joint endpoint of diameter they must be of equal radii and in any case the above method will get you result.
| {
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Understanding concatenating the empty set to any set. I know that concatenating the empty set to any set yields the empty set. So, $A \circ \varnothing = \varnothing$. Here $A$ is a set of strings and the concatenation ($\circ$) of two sets of strings, $X$ and $Y$ is the set consisting of all strings of the form $xy$ w... | The wrong line of thinking is almost correct, in the following way. Let $\epsilon$ be the empty string. For any string $a$, we have $a\epsilon=a$. Let $\{\epsilon\}$ be the set containing exactly one element, namely the empty string. Then for any set of strings $A$, we have $A\circ\{\epsilon\}=A$.
The real issue, then,... | {
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A set of basic abstract algebra exercises I wanted to review some basic abstract algebra. Here's a few problems for which I am seeking solution verification. Thank you very much in advance!
$\textbf{Problem:}$ Let $H$ be a subgroup of $G$, and let $X$ denote the set of all the left cosets of $H$ in $G$. For each elemen... | It looks good in its current state, but some of the arguments could be simplified. Do recall that
$$xH = yH \iff x^{-1} y \in H$$
So for the proof of the first one, we have
\begin{align}
\rho_\alpha (xH) = \rho_{\alpha}(x'H) &\iff (\alpha x) H = (\alpha x') H \\
&\iff (\alpha x)^{-1} (\alpha x') \in H \\
&\iff x^{-1} \... | {
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Why doesn't this set have a supremum in a non-complete field? Why doesn't the set $\{x:x^2<5\}$ have a supremum in $\mathbb{Q}$? I know that the rational numbers aren't a complete field, but I'm still not understanding how a set can have upper bounds, but no least upper bound in a field.
In $\mathbb{Z}$ for example, $\... | Suppose $q$ is a rational number which is an upper bound for your set. Since $\sqrt{5}$ is irrational, it must be the case that $q>\sqrt{5}$. But then there exists a rational number $q'$ such that
$$\sqrt{5} < q' < q$$
so $q$ couldn't have been a least upper bound.
Added: if you want to work entirely inside $\mathbb{Q... | {
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Entire Function Problem in Complex Analysis I am currently working on some review problems in complex analysis and came upon the following conundrum of a problem.
"If $f(z)$ is an entire function, and satisfies $|f(z^2)|\le|f(z)|^2$, prove that f(z) is a polynomial."
My intuition tells me to show that f(z) has a pole a... | Let
$$
M=\sup_{|z|=2}|f(z)|
$$
Then, with the condition given, it can be proven inductively that
$$
\sup_{|z|=2^{2^n}}|f(z)|\le M^{2^n}
$$
which implies
$$
|f(z)|\le|z|^{2\log_2(M)}
$$
We can use Cauchy's Theorem to give
$$
f^{(n)}(z)=n!\int_{\gamma_R}\frac{f(w)\,\mathrm{d}w}{(w-z)^{n+1}}
$$
where $\gamma_R$ is the cir... | {
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Convergence implies lim sup = lim inf Could someone please explain to me how the following can be proven? I get the intution but don't know how to write it rigorously.
Thank you.
| Both $limsup$ and $liminf$ are limit points of a sequence; the largest and smallest limit points respectively of a sequence, just as $lim$ is a limit point . Assuming your space is Hausdorff, or something else that guarantees that the limit is unique, then there is only one limit point, so we must have $liminf=limsup=... | {
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General formula needed for this product rule expression (differential operator) Let $D_i^t$, $D_i^0$ for $i=1,\dots,n$ be differential operators. (For example $D_1^t = D_x^t$, $D_2^t = D_y^t,\dots$, where $x$, $y$ are the coordinates).
Suppose I am given the identity
$${D}_a^t (F_t u) = \sum_{j=1}^n F_t({D}_j^0 u){D}_a... | This is probably better served as a comment, but I can't add one because of lack of reputation. If I understand correctly you are probably looking for the multivariate version of Faà di Bruno's formula, beware that the Wikipedia entry uses slightly different notation.
| {
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What is the "reverse" of the cartesian product? Suppose $A = \{a_1,a_2 \}$ and $B = \{b_1,b_2 \}$. Then $A \times B = \{(a_1,b_1), (a_1,b_2), (a_2,b_1), (a_2,b_2) \}$. What is the "reverse" of this operation? In particular, what would $A \div B$ be?
The motivation for this question is from relational algebra. Consider ... | It looks like you want to define it as follows:
Given sets $X,Y,$ $A\subseteq X\times Y,$ and $B\subseteq Y$ we define $$A\div B:=\bigl\{x\in X\mid\{x\}\times B\subseteq A\bigr\}.$$
As far as I'm aware, there is no standard name for this.
More generally, if you wanted it to work for arbitrary sets (not just subsets o... | {
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Looking for a source: Fourier inversion of $f \in L^1$ Is there a book where I can find a thorough proof of the following assertion?
Let $f \in L^1(\mathbb{R}^d)$ be continuous at zero and $\hat{f}\ge0$. Then $\hat{f} \in L^1(\mathbb{R}^d)$ and
$$f(t) = \int_{\mathbb{R}^d} \hat{f}(\xi)e^{2\pi i\, t\xi} \,d\xi$$
al... | In terms of the proof please see here: Proof of Fourier Inverse formula for $L^1$ case
Further to another German reference that rather pays attention to context than proof is here http://www.math.ethz.ch/education/bachelor/seminars/hs2007/harm-analysis/FT2.pdf
| {
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Prove that $f(x)=\int_{0}^x \cos^4 t\, dt\implies f(x+\pi)=f(x)+f(\pi)$ How to prove that if $f(x)=\int_{0}^x \cos^4 t\, dt$ then $f(x+\pi)$ equals to $f(x)+f(\pi)$
I thought to first integrate $\cos^4 t$ but this might be the problem of Definite Integral
| $$f(x+\pi)-f(\pi)=\int_0^{x+\pi}\cos^4t dt-\int_0^{\pi}\cos^4t dt=\int_{\pi}^{x+
\pi}\cos^4t dt$$
Putting $u=t-\pi, dt=du$ and $\cos t=\cos(u+\pi)=-\cos u$
$$\int_{\pi}^{x+\pi}\cos^4t dt=\int_0^x(-\cos u)^4du=f(x)$$
| {
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I want to show that ker f is a normal subgroup of some group $X$ Suppose I have two groups, call them $X$ and $Y$, and I let $f : X \longrightarrow Y$ be a group homomorphism. I want to prove that the ker $f$ is a normal subgroup of $X$. Here is my attempt. Let me know how my proof looks and if I am missing details:
We... | Yes, it looks entirely fine. You might want to explicitly state what $I$ means, but the logic is sound.
| {
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Prove the Following Property of an Ultrafilter In her text Introduction to Modern Set Theory , Judith Roitman defined a filter of a set $X$ as a family $F$ of subsets of $X$ so that:
(a) If $A \in F$ and $X \supseteq B \supseteq A$ then $B \in F$.
(b) If $A_1, ... ,A_n$ are elements of $F$, so is $A1 \cap ... \cap A... | If I were writing such text, I would have pointed it out in order to remind the reader of this. Especially when in just one sentence we derived a contradiction from it.
| {
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How to find the integral by changing the coordinates? Let R be the region in the first quadrant where
$$3 \geq y-x \geq 0$$
$$5 \geq xy \geq2$$
Compute
$$\int_A (x^2-y^2)\,dx\,dy.$$
I tried to use $ u= y-x, v= xy$ as my change of coordinates, but then I don't know how to solve it.
Can someone help me?
| For the Jacobian, use this fact that:
$$\frac{\partial(x,y)}{\partial(u,v)}=\frac{1}{\frac{\partial(u,v)}{\partial(x,y)}}$$ provided $\frac{\partial(x,y)}{\partial(u,v)}\neq 0$.
| {
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"timestamp": "2023-03-29T00:00:00",
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how to find out any digit of any irrational number? We know that irrational number has not periodic digits of finite number as rational number.
All this means that we can find out which digit exist in any position of rational number.
But what about non-rational or irrational numbers?
For example:
How to find out which ... | You can use continued fraction approximations to find rational numbers arbitrarily close to any irrational number.
For $\sqrt 2$ this is equivalent to the chain of approximations $\frac 11, \frac 32, \frac 75, \frac {12}{17} \dots$ where the fraction $\cfrac {a_{n+1}}{b_{n+1}}=\cfrac {a_n+2b_n}{a_n+b_n}.$
The accuracy ... | {
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Commuting an $\int$ improper at its both ends and $\lim$ I am working on the following problem:
Let $f, g$ be continuous nonnegative functions defined and improperly-integrable on $(0, \infty)$ Furthermore, assume they satisfy
$$
\lim_{x\rightarrow 0}f(x) = 0 \wedge \lim_{x\rightarrow\infty}xg(x)=0.
$$
Then prove ... | Here is a way to use a direct bound for looking at the integral over $[1,\infty)$:
$\int_1^\infty f(x) ng(nx) dx \leq \int_n^\infty f(u/n) g(u) du = \int_n^\infty \frac{f(u/n)}{u} (u g(u)) du \leq \frac{1}{n} \int_n^\infty f(u/n) (ug(u)) du$
The latter inequality follows from the fact that in the integrand $u > n \ge... | {
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Field structure on $\mathbb{R}^2$ I have the following question:
Is there a simple way to see that if we put a multiplication $*$ on $\mathbb{R}^2$ (considered as a vector space over $\mathbb{R}$) such that with usual addition and this multiplication $\mathbb{R}^2$ becomes a field, then there exists a nonzero $(x,y)$ s... | Sorry if I make it too elementary: If $1\in\mathbb R^2$ denotes $1$ of your field, and if $x\in\mathbb R^2$ is not its real multiple: $1,x,x^2$ are linearly dependent (over $\mathbb R$), i.e. $ax^2+bx+c=0$ for some $a,b,c$, and $a\neq 0$ (as $x$ is not a multiple of $1$), so we can suppose it's 1. If we complete squa... | {
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Limit point intuition Quoting Rudin,
"A point $p$ is a limit point of the set $E$ if every neighborhood of $p$ contains a point $q\not=p : q \in E$."
This would imply that the points in an open ball would all be limit points, since for any $p$ in $E$ there are $q$ such that $d(p,q) < r$ for all $q \in E$. So E is also ... | You can think of the set of limit points $L(S)$ of a set $S$ as all points which are "close to" $S$. In the example of an open ball in $\mathbb{R}^n$, the limit points are all points of the open ball, plus all points lying on the boundary, since every punctured neighborhood of such points will intersect the set.
Note,... | {
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Radius and amplitude of kernel for Simplex noise I'm wondering if formulas exist for the radius and amplitude of the hypersphere kernel used in Simplex noise, generalized to an arbitrary number of dimensions. Ideally I'd like an answer with two equations in terms of n (number of dimensions) that give me r (radius) and ... | The formula for the radius $r$ is simple.
$$r^2 = \frac {1} {2}$$
This holds for all values of $N$. Let me explain.
The simplex noise kernel summation radius $r$ should be the height of the N-simplex. If the kernel summation radius is larger than this, the kernel contribution will extend outside of the simplex. This... | {
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What are better words to use in an article than "obvious"? I've heard often than it is ill-form to use the word "obvious" in a research paper. I was hoping to gather a list of less offensive words that mean generally the same thing.
For example, one that I can think of is the word "direct".
So instead of saying "...o... | evidently, visibly, naturally, undeniably..
Words like trivially, and obviously sound disrespectful, it is as if the author is mocking the reader.
Also they sound 'empty' and many authors use these words to make up for the incompleteness in their work.
Mathematics is about deduction, not intuition. So any word that d... | {
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How to find asymptotes of implicit function? How to find the asymptotes of the implicit function $$8x^3+y^3-6xy-3=0?$$
| I have seen you are interested in doing problems by Maple so, the following codes may help you machineary:
[> f:=8*x^3+y^3-6*x*y-3:
t := solve(f = 0, y):
m := floor(limit(t[1]/x, x = -infinity));
$$\color{blue}{m=-2}$$
[> h:=floor(limit(t[1]-m*x, x = -infinity));
$$\color{blue}{h=-1}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/474860",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
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Diagonalizable matrix with only one eigenvalue I have a question from a test I solved (without that question.. =)
"If a matrix $A$ s.t $A$ is in $M(\mathbb{C})$ have only $1$ eigenvalue than $A$ is a diagonalizable matrix"
That is a false assumption since a ($n\times n$ matrix) a square matrix needs to have at least $... | You need your $n\times n$ matrix to have n linearly-independent eigenvectors. And the identity matrix is already in diagonal form.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/474939",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 2
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Leibniz notation for high-order derivatives What is the reason for the positioning of the superscript $n$ in an $n$-order derivative $\frac{d^ny}{dx^n}$? Is it just a convention or does it have some mathematical meaning?
| Several people have already posted answers saying it's $\left(\dfrac{d}{dx}\right)^n y$, so instead of saying more about that I will mention another aspect.
Say $y$ is in meters and $x$ is in seconds; then in what units is $\dfrac{dy}{dx}$ measured? The unit is $\text{meter}/\text{second}$. The infinitely small quant... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/475016",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 3
} |
Issues in calculating the gradient I am trying to calculate the gradient of a certain expression. I am not sure if it's possible. I have the following
$f(\alpha_1,\alpha_2,\Lambda) = \log(|2Q_1+2Q_2 +2Q_3|)$
$Q_1$ is a diagonal matrix with the diagonal terms equal to $\alpha_1$
$Q_2$ is a diagonal matrix with the diago... | Well, if you are interested in finding gradient with the respect to the parameters ($\alpha_1, \alpha_2, \Lambda_{ii}$) separately (without concatenating them in one vector and searching the derivative wrt a vector) you can use some matrix calculus identities (but first you can pull out the factor of $2$ out of the log... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/475143",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
$R/I$ when $R$ is the ring of real continuous functions If $R$ is the ring of all real continuous functions on $[0,1]$, I am trying to find $R/I$ where
$$I=\{f\in{R}|f(.5)=0\}$$
Showing $I$ is an ideal is not a problem since we're defining addition and multiplication as $$(f+g)(x)=f(x)+g(x).$$
$$(fg)(x)=f(x)g(x)$$but ... | Hint: What is the kernel of the ring homomorphism $R\to \mathbb R$, $f\mapsto f(0.5)$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/475241",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Please Explain $\lg(T(N)) = 3 \lg N + \lg a$ is equivalent to $ T(N) = aN^3$ I'm reading Algorithms by Kevin Wayne and Robert Sedgewick.
They state that:
$\lg(T(N)) = 3 \lg N + \lg a $
(where $a$ is constant) is equivalent to
$T(N) = aN^3$
I know that $\lg$ means a base $10$ logarithm and that $\lg(T(N))$ means the i... | Simply raise $10$ to the power of both sides of the equation:
$\large{10^{\log {T(N)}}=10^{3\log N +\log a}=10^{3\log N}\cdot10^{\log a}=(10^{log N})^3\cdot10^{\log a}}$
Since by definition $\log b = c \iff 10^c=b$, it follows that $10^{\log b}=b$, and thus
$T(N)=N^3 \cdot a$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/475281",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Rational function with absolute value $1$ on unit circle
What is the general form of a rational function which has absolute value $1$ on the circle $|z|=1$? In particular, how are the zeros and poles related to each other?
So, write $R(z)=\dfrac{P(z)}{Q(z)}$, where $P,Q$ are polynomials in $z$. The condition specifie... | Answer your question about why $M$ is constant: it's simply because $M$ is a quotient of two polynomials. If the quotient is $1$ on the unit circle, it means these two polynomials are equal at all the points of the circle. This implies that these two polynomials are the same. So $M$ is identically $1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/475344",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 5,
"answer_id": 2
} |
Generating function for $\sum_{k\geq 1} H^{(k)}_n x^ k $ Is there a generating function for
$$\tag{1}\sum_{k\geq 1} H^{(k)}_n x^ k $$
I know that
$$\tag{2}\sum_{n\geq 1} H^{(k)}_n x^n= \frac{\operatorname{Li}_k(x)}{1-x} $$
But notice in (1) the fixed $n$.
| Let $\psi(x)=\frac{\Gamma'}{\Gamma}(x)=\frac{d}{dx}\log\Gamma(x)$ be the digamma function. For $N$ a positive integer, we have
$$
\psi(x+N)-\psi(x)=\sum_{j=0}^{N-1}\frac{1}{x+j}
$$
(this follows from $x\Gamma(x)=\Gamma(x+1)$ and induction).
Now
\begin{eqnarray*}
\sum_{k\geq 1}H_n^{(k)}x^k&=&\sum_{k\geq 1}\sum_{j=1}^n \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/475491",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Schreier generators I am facing some problem in understanding the proof of the following theorem. Can somebody provide me a simple proof .
Given $G=\langle A \rangle$ and $H \leq G$, and $R$ is the coset representatives for $H$ in $G$.
Let $B=\{r_1ar^{-1}_2 | r_1,r_2 \in R, a \in A\}\cap H.$
Then $B$ generates $H$.
| Your assertion is a few incorrect. It must be as follows:
Given $G=\langle A \rangle$ and $H\le G$, and $R$ is a set of representatives of the right cosets for H in G. Let B={r1ar−12|r1,r2∈R,a∈A}∩H. Then B generates H. Let $B = \{r_1ar^{-1}_2 | r_1 \in R, a \in A\}$, where $r_2$ is the representative of the coset... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/475590",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that in a parabola the tangent at one end of a focal chord is parallel to the normal at the other end.
Prove that in a parabola the tangent at one end of a focal chord is
parallel to the normal at the other end.
Now, I know prove this algebraically, and that's very easy, but I am not getting any visual pictur... | Here's a geometric proof, based on the fact that a line (thought of as a light ray) going through the focus of a parabola reflects to a line parallel to the axis of the parabola. This is sometimes called the reflective property of the parabola. Call the focus $F$, and have the parabola arranged with its axis the $y$ ax... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/475666",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
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Evaluating Laplace Transform I have a Laplace transform function of the following form and I'm trying to evaluate it. From my research I think I need to take the Inverse Laplace Transform and then integrate, but I'm having trouble working that out, or if that's even what I have to do.
$$\frac{s^2}{s^2+10s+25} $$
and
$$... | The definition of a Laplace transform leads to the following expression for the inverse Laplace transform of a function $F(s)$:
$$f(t) = \frac{1}{i 2 \pi} \int_{c-i \infty}^{c+i \infty} ds \, F(s) \, e^{s t}$$
where $c$ is a real number larger than the real parts of all poles of $F$ in the complex $s$ plane. That is, ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Can $P \implies Q$ be represented by $P \vee \lnot Q $? Source: p 46, How to Prove It by Daniel Velleman
Though the author writes $Q$ (the original apodosis) as 'You'll fail the course',
for brevity I shorten $Q$ to 'You fail'.
Let $P$ be the statement “You will
neglect your homework” and $Q$ be “You fail.”
Then “... | This follows simply from the Law of Excluded Middle: $P \lor \neg P$ for all propositions $P$.
Let's assume $P \rightarrow Q$ and deduce $Q \lor \neg P$.
By the Law of Excluded Middle, we have either $P$ or we have $\neg P$. We do a case analysis over which one is true:
If we got a $P$, by our assumption, we can dedu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/475776",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 4
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Is the number 8 special in turning a sphere inside out? So after watching the famous video on youtube How to turn a sphere inside out I noticed that the sphere is deformed into 8 bulges in the process. Is there something special about the number 8 here? Could this be done with any number of bulges, including 2?
Image:
... | No, 8 isn't special beyond it being the choice they made for that specific video.
The software the group wrote to make that video allowed you to choose that parameter arbitrarily. I bet if you spent some time digging you could find that software somewhere on the internet, and create your own eversion videos with a di... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/475837",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 1,
"answer_id": 0
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Convergence of $\{nz^n\}_1^{\infty}.$
Discuss completely the convergence and uniform convergence of the sequence $\{nz^n\}_1^{\infty}.$
If $|z|\geq 1$, then $|nz^n|=n|z|^n\geq n$ diverges, so the sequence $nz^n$ also diverges.
If $|z|<1$, it should converge to $0$. So for any $\varepsilon$, we must find $N$ such that... | Hint: From real analysis/calculus you may recall the result
$$
\lim_{n\to\infty}\frac{n}{a^n}=0,
$$
whenever the constant $a>1$. An exponential function grows faster than a power function or some catch-phrase like that is sometimes associated with this result.
Fix a constant $a>1$ and consider the numbers $z$ such that... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/475907",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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At what angle do these curves cut one another? I'm working on an exercise that asks this: At what angle do the curves $$y = 3.5x^2 + 2$$ and $$y = x^2 - 5x + 9.5$$ cut one another? I have set these equations equal to one another to find two values for x. Namely, $x = 1$ and $x = -3$ as intersections. How should I proce... | You know the curves cut themselves at $x=1$ and $x=-3$. Let's consider a general case you might find helpful. Consider two functions $f,g$ that intersect at a point $x=\xi$.
Consider now the tangent line of $f$ at $x=\xi$. What angles does it make with the $x$ axis? It shouldn't be new news that $\tan\theta=f'(\xi)$. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/475951",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Taking the limit of $n(e^{-1/n}-1)$ as $n$ approaches infinity The form is infinity times zero and that is indeterminate which means I need to use L'Hospital's rule, but I have tried to do that but every time I would find another indeterminate form. How can I use sneaky algebra or sneaky replacements to find the answe... | L'Hospital's Rule is not my favourite approach, since a computation replaces insight about the behaviour of the function. But one cannot deny its usefulness. We do two L'Hospital's Rule calculations.
Calculation 1: We want to find
$$\lim_{n\to\infty}\frac{e^{-1/n}-1}{1/n}.\tag{1}$$
Let $x=\frac{1}{n}$. As $n\to\infty$,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/476025",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Possible Research Topics for High School I am a highschool student some experience with Math Olympiads and I will be taking a Scientific Research class next year. I would like to ask for interesting Mathematics topics that I could consider - I have tried going online for possible research topics but I couldn't determin... | I'm a fan of mathematical games for projects like these. They are fun, don't (necessarily) require advanced math -- I 've taught basic theory to groups 10 year olds -- and there are lots of open problems. Check out Winning Ways, by Berlekamp, Conway, and Guy, and M. Albert, R. J. Nowakowski, D. Wolfe, Lessons in Play.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/476154",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Prove $X^2+Y^2-1$ is irreducible using geometrical tools. I'm trying to understand what is meant in this paragraph:
of "Conics and Cubics. A Concrete Introduction to Algebraic Curves (by Robert Byx)":
He wants to prove that the polynomial $X^2+Y^2-1$ is irreducible using geometrical tools. I have the following doub... | *
*A line in $k^2$ has the form $aX+bY+c=0$ where $a$ and $b$ cannot both be zero. We may assume without loss of generality that $a\ne 0$, so that we have $X=-a^{-1}bY-a^{-1}c$. It follows that for any $\alpha\in k$, the point
$$(-a^{-1}b\alpha-a^{-1}c,\alpha)$$
lies on the line. It follows that the number of point... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/476218",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
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If $a>0$, $b>0$ and $n\in \mathbb{N}$, show that $aIf $a>0$, $b>0$ and $n\in \mathbb{N}$, show that $a<b$ if and only if $a^n<b^n$. Hint: Use mathematical induction.
Having trouble with the proof that if $a<b$ then $a^n<b^n$.
So far I have;
Assume $a<b$
then $a^k<b^k$ for $k=1$
Assume $\exists m \in \mathbb{n}$ such t... | Hint: Do it in two steps, $a^{m+1}=a^m a\lt a^m b\lt b^mb=b^{m+1}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/476287",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
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How prove this geometry $\Delta PCA \sim\Delta PBD$ let the circle $O_{1} $ and the circle $O_{2}$ the radius of is $r_{1},r_{2}$ respectively,and the circle $O_{1}$and $O_{2}$ intersection with $A$ and $B$,and the tangent to $O_{1}$ at $C$,and the tangent to $O_{2}$ at $D$, and such
$\dfrac{PC}{PD}=\dfrac{r_{1}}{r_... | Here's a partial solution.
I'll revise notation a little bit and use coordinates to help set the stage. My circles have centers $H(-h,0)$ and $K(k,0)$ and respective radii $r$ and $s$. (Without loss of generality, we assume $r > s$; the case $r=s$ is left to the reader.) The circles meet at points $A(0,a)$ and $B(0,-a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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A variant of the Schwartz–Zippel lemma Let $f \in \mathbb{F}[x_1,\ldots,x_n]$ be a nonzero polynomial. Let $d_1$ be the maximum exponent of $x_1$ in $f$ and let $f_1$ be the coefficient of $x_1^{d_1}$ in $f.$ Let $d_2$ be the maximal exponent of $x_2$ in $f_1$ and so on for $d_3,\ldots,d_n.$
I would like to show that i... | This "variant of the Schwartz-Zippel Lemma" is in fact Lemma 1 from Jack Schwartz's original paper.
Note that although it looks nice to phrase the statement and/or the proof in probabilistic language, it is certainly not necessary to do so: Schwartz phrases it as a pure counting argument, which is (even) shorter than J... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Partial fractions for inverse laplace transform I have the following function for which I need to find the inverse laplace transform:
$$\frac1{s(s^2+1)^2}$$
Am I correct in saying the partial fraction is:
$$\frac1{s(s^2+1)^2}=\frac{A}{s}+\frac{Bs+C}{s^2+1}+\frac{Ds+E}{(s^2+1)^2}$$
| Another way to evaluate the ILT when there are just poles (i.e., no branch points like roots and logarithms) is to apply the residue theorem. In this case, the ILT is simply the sum of the residues at the poles of the LT. That is,
$$\begin{align}f(t) &= \operatorname*{Res}_{s=0} \frac{e^{s t}}{s(1+s^2)^2}+ \operatorn... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/476505",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Does a logarithmic branch point imply logarithmic behavior? The complex logarithm $L(z)$ is given by $$L(z)=\ln(r)+i\theta$$ where $z=re^{i\theta}$ and $\ln(x)$ is the real natural logarithm. It is well known that $L(z)$ then sends each $z$ to infinitely many values, each of which are different by an integer multiple o... | I think I may have an answer;
$f(z)=z^p$ with $p$ an irrational number. If $p$ is rational, then the order of the branch point $f(z)$ at $z=0$ is just the denominator in lowest terms. So if we take the limit as $p$ goes to an irrational number, we should get an infinite order branch point.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/476568",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
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How many times the parabola $y=x^2$ intersects the origin? The answer of this question is 1, right? but I'm about to study algebraic curves by this book
and I was surprised by this theorem:
If I'm right the immediate corollary of that is $y=x^2$ intersects the origin two times.
Then he continues given an example:
I... | You have to be a bit careful there. If we want to find the intersection of $y=x^2$ and the line $y=0$ within this framework, we would (or, at least, could) say
$$
y = p(x) = x^2\\
g(x,y) = y-x^2
$$
The theorem then tells us that as long as $\mathbf{y-p(x)}$ is not a factor of $\mathbf{g(x,y)}$, there are two intersecti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/476641",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Simple Algorithm Running Time Analysis
A sorting algorithm takes $1$ second to sort $1,000$ items on your local machine. How long will it take to sort $10,000$ items
*
*if you believe that the algorithm takes time proportional to $n^2$,
and
*if you believe that the algorithm takes time roughly proport... |
Yes, it seems correct. -Servaes
| {
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"url": "https://math.stackexchange.com/questions/476701",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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When does L'Hopital's rule pick up asymptotics? I'm taking a graduate economics course this semester. One of the homework questions asks:
Let $$u(c,\theta) = \frac{c^{1-\theta}}{1-\theta}.$$ Show that $\lim_{\theta\to 1} u(c) = \ln(c)$. Hint: Use L'Hopital's rule.
Strictly speaking, one can't use L'Hopital's rule; at... | One can imagine a situation in which L'Hospital's Rule does not apply, but gives the right answer. This is not one of them. The limit is not $\ln c$. A glance at the expression shows that the limit from the left is "$\infty$" and the limit from the right is "$-\infty$."
Remark: Suppose that for some constants $a,b,c,d$... | {
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"timestamp": "2023-03-29T00:00:00",
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Proving the real numbers are complete In Rudin's book, the following proof is published:
Let $A$ be the set of all positive rationals $p : p^2 < 2$. Let $B$ be the set of all positive rationals $p : p^2 > 2$. $A$ contains no largest number and $B$ contains no smallest.
Let q be a rational. More explicitly, $$\forall p... | if $p^2<2$
let's add 2p to both sides of the inequality
then $p^2+2p<2+2p$ then $p(p+2)<2(p+1)$ then $p<\frac{2(p+1)}{p+2}$ so let's set $q=\frac{2(p+1)}{p+2}$ then we have if $p^2<2$ then $p<q$. what is left to show is that $q^2<2$
and that would show that $0<p<q$ and $q \in A$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/476812",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 6,
"answer_id": 5
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The intuition behind Trig substitutions in calculus I'm going through the MIT open calculus course, and in one of the lectures (19-28min marks) the professor uses the trig substitution $x = \tan \theta$ to find the integral of $\frac{dx}{x^2 \sqrt{1+x^2}}$.
His answer: $-\csc(\arctan x) + c$, which he shows is equival... | Your method is based on the identity
$$
1+\tan ^{2}\theta =\sec ^{2}\theta .
$$
But there is another standard method to integrate by substitution an irrational
function of the type $f(R(x),\sqrt{a^2+x^{2}})$, where $R(x)$ is a rational function of $x$. This alternative method, which is based on the identity
$$1+\sinh... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/476890",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
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Complex power of a complex number Can someone explain to me, step by step, how to calculate all infinite values of, say,
$(1+i)^{3+4i}$?
I know how to calculate the principal value, but not how to get all infinite values...and I'm not sure how to insert the portion that gives me the other infinity values.
| Let's suppose you've already defined $\log r$ for real $r > 0$, say, using Taylor series. Then given $z, \alpha \in \mathbb{C}$, you can define
$$z^{\alpha} = \exp(\alpha \log z)$$
where
$$\exp(w) = \displaystyle \sum_{j=0}^{\infty} \dfrac{z^j}{j!} \qquad \text{and} \qquad \log(w) = \log |w| + i \arg(w)$$
This is not w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/476968",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 5,
"answer_id": 1
} |
A question regarding a step in power method justification (Writing a vector in terms of the eigenvectors of a matrix) Let $A$ be a $t \times t$ matrix. Can we present any $t \times 1$ vector, as a linear combination of eigenvectors of $A$?
I think this should not be the case unless all eigenvectors of $A$ happened to b... | As @oldrinb already said, eigenvectors corresponding to distinct eigenvalues are always linearly independent.
Next, a matrix has a basis of eigenvectors if and only if it's diagonalisable, which is not always the case, consider $$\begin{pmatrix}1&1\\0&1\end{pmatrix}.$$
This matrix has only one eigenvector (up to a cons... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/477037",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Asymptotic correlation between sample mean and sample median Suppose $X_1,X_2,\cdots$ are i.i.d. $N(\mu,1)$. Show that the asymptotic correlation between sample mean and sample median (after suitably centering and renormalization) is $\sqrt{\frac{2}{\pi}}$.
| Obtain
this paper, written by T.S. Ferguson, a professor at UCLA (his page is here).
It derives the joint asymptotic distribution for the sample mean and sample median.
To be specific, let $\hat X_n$ be the sample mean and $\mu$ the population mean, $Y_n$ be the sample median and $\mathbb v$ the population median. Let... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/477115",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
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Equations for double etale covers of the hyperelliptic curve $y^2 = x^5+1$ Let $X$ be the (smooth projective model) of the hyperelliptic curve $y^2=x^5+1$ over $\mathbf C$.
Can we "easily" write down equations for all double unramified covers of $X$?
Topologically, these covers correspond to (normal) subgroups of index... | Let me elaborate a bit my comment.
This part works for any smooth projective curve $X$ in characteristic $\ne 2$. A double cover $Y\to X$ is given, as you said, by a quadratic extension $L$ of $K={\mathbf C}(X)$. It is an elementary result that such an extension is always given by adjoining a square root $z$ of some $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/477176",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 2,
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Divisibility of sequence Let the sequence $x_n$ be defined by $x_1=1,\,x_{n+1}=x_n+x_{[(n+1)/2]},$ where $[x]$ is the integer part of a real number $x$. This is A033485. How to prove or disprove that 4 is not a divisor of any its term? The problem belongs to math folklore. As far as I know it, M. Kontsevich authors tha... | For $n\in\Bbb Z^+$ let $u_n=x_n\bmod 4$, and let $\oplus$ denote addition modulo $4$. We have the recurrences
$$\left\{\begin{align*}
u_{2n}&=u_{2n-1}\oplus u_n\\
u_{2n+1}&=u_{2n}\oplus u_n\;,
\end{align*}\right.$$
and the first few values are $u_1=1,u_2=2,u_3=3$, and $u_4=1$. The desired result is an immediate corolla... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/477257",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Expected overlap Suppose I have an interval of length $x$ and I want to drop $n$ sticks of unit length onto it (where $\sqrt x<n<x$). What is the expected overlap between sticks? ($x$ can be assumed to be large enough that edge effects are negligible.)
I assume this is a standard problem and has a name but I don't know... | Let $t \in [0,x]$. The probability that a given stick hits $t$ (assuming left end-point of stick chosen uniformly in $[0,x-1]$) is
$p(t) = \begin{cases} \frac{t}{x-1} & t < 1 \\ \frac{1}{x-1} & 1 < t < x-1 \\ \frac{x-t}{x-1} & x-1 < t < x\end{cases}$.
The probability that $\geq 2$ sticks hit $t$ is (via the compleme... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/477320",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
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I need to calculate $x^{50}$ $x=\begin{pmatrix}1&0&0\\1&0&1\\0&1&0\end{pmatrix}$, I need to calculate $x^{50}$
Could anyone tell me how to proceed?
Thank you.
| The Jordan Decomposition yields
$$
\left[
\begin{array}{r}
1 & 0 & 0 \\
1 & 0 & 1 \\
0 & 1 & 0
\end{array}
\right]
=
\left[
\begin{array}{r}
0 & 0 & 2 \\
-1 & 1 & 1 \\
1 & 1 & 0
\end{array}
\right]
\left[
\begin{array}{r}
-1 & 0 & 0 \\
0 & 1 & 1 \\
0 & 0 & 1
\end{array}
\right]
\left[
\begin{array}{r}
0 & 0 &... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/477382",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 8,
"answer_id": 5
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Proof of properties of injective and surjective functions. I'd like to see if these proofs are correct/have them critiqued.
Let $g: A \to B$ and $f: B \to C$ be functions. Then:
(a) If $g$ and $f$ are one-to-one, then $f \circ g$ is one-to-one.
(b) If $g$ and $f$ are onto, then $f \circ g$ is onto.
(c) If $f \c... | Nicely done! In part (f), I think you mean that $g$ is not surjective since there is nothing in $A$ that $g$ will take to $2,$ but the idea is spot on.
A minor critique for your proof of (b): I would instead suggest that you take an arbitrary $c\in C$, use surjectivity of $f$ to conclude that there is some $b\in B$ suc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/477453",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "24",
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Region in complex plane with $|1-z|\leq M(1-|z|)$ Let $M>0$. Describe the region in the complex plane such that $|1-z|\leq M(1-|z|)$.
To start, I take $M=1$. The inequality becomes $|1-z|\leq 1-|z|$. But by triangle inequality, we have $|1-z|+|z|\geq |(1-z)+z| = 1$. We must have equality, and it holds when $z\in [0,1]$... | Your region consists of all $z \in \mathbb C$, such that the ratio $$\frac{|1-z|}{1-|z|} $$ is bounded (by $M$). As you mentioned only points within the unit disk are admissible.
More precisely, the region is a subset of the unit disk, which is contained within a circular wedge of angle $\alpha=\alpha(M)$ (the Stolz an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/477539",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
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Convergence of $\sum_{n=0}^{\infty}\frac{z^n}{1+z^{2n}}$
For what complex values of $z$ is $$\sum_{n=0}^{\infty}\dfrac{z^n}{1+z^{2n}}$$ convergent?
I would like to write the sum as a power series, because with a power series we can determine the radius of convergence. But in this case it seems untidy. We have $\dfrac... | Hint: When $|z|<1$ we have $$\left|\frac{z^n}{1+z^{2n}}\right| \le \frac{|z|^n}{1-|z|^{2}},$$
when $|z|>1$ we have
$$\left|\frac{z^n}{1+z^{2n}}\right| \sim {|z|^{-n}}$$
and when $|z|=1$ we have
$$\left|\frac{z^n}{1+z^{2n}}\right| \not\to 0.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/477610",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Interesting question in analysis I am trying to prove this :
Consider $\Omega \subset R^n$ ( $n \geq 2$) a bounded and open set and $u $ a smooth function defined in $\overline{\Omega}$. Suppose that $u(y) = 0$ for $y \in \partial \Omega$ and suppose that exists a $\alpha >0$ such that $|\nabla u (x)| = \sqrt{\displays... | I think there is something wrong here.
If $u$ is zero in $\partial \Omega$, then $u$ has a minimum or a maximum in $\Omega$. So there is a point $P$ in $\Omega$ such that $\nabla u(P)=0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/477749",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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General question about 'vieta jumping' Suppose I want to prove that a variable posesses a certain property (e.g. is a square). For example if I wanted to prove that $x$ in $\frac{x^2+y^2+1}{xy} = k$ has the property of being a square (It is obviously false, but suppose it is true ). Is it then necessary to fix $x$ and ... | In this case the problem is very simple you just need to use Vieta Jumping and find the smallest solution. In other problems that uses vieta's jumping you normally fix one varible an then you find the possible k. Here you have an example http://www.artofproblemsolving.com/community/c6h339649
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/477830",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
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$h(x)={f(x)\over x}$ is decreasing or increasing or both over $[0,\infty)$ $f$ is real valued function on $[0,\infty)$ such that $f''(x)>0$ for all $x$ and $f(0)=0$
Then $h(x)={f(x)\over x}$ is decreasing or increasing or both over $[0,\infty)$
$h'(x)={xf'(x)-f(x)\over x^2}$
What I can conclude from here?
| Since f''(x) > 0 on [0,∞] we have that $\int f''(x)\,dx$ = f'(x) > 0 for all x ≥0. So f is increasing everywhere (we know that anyway since f''(x) > 0 means f is concave upwards, but maybe this is a little more precise). Further it is increasing faster than x, because f'(x) = 1 and f''(x) = 0. So the numerator of f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/477889",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Arithmetical Functions Sum, $\sum\limits_{d|n}\sigma(d)\phi(\frac{n}{d})$ and $\sum\limits_{d|n}\tau(d)\phi(\frac{n}{d})$ $$\sum_{d|n}\sigma(d)\phi\left(\frac{n}{d}\right)=n\tau(n)
,\\ \sum_{d|n}\tau(d)\phi\left(\frac{n}{d}\right)=\sigma(n)$$
The problem (7.4.15) of Burton's Elementary Number Theory has been request to... | Prove that both sides are multiplicative functions and that they coincide when $n$ is a prime power.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/477961",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 2
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Fast Matlab Code for hypergeometric function $_2F_1$ I am looking for a good numerical algorithm to evaluate the hypergeometric function $_2F_1$ in Matlab (hypergeom in Matlab is very slow). I looked across the Internet, but did not find anything useful. Does anybody here have an idea where to find a fast algorithm to ... | There is Matlab source code for J. Pearson's master's thesis "Computation of Hypergeometric Functions". The thesis is a available as
http://people.maths.ox.ac.uk/porterm/research/pearson_final.pdf
and the Matlab code URL is http://people.maths.ox.ac.uk/porterm/research/hypergeometricpackage.zip (I cannot judge the cod... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/478052",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 0
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Wilson's theorem intuition Wilson's Theorem: $p$ is prime $\iff$ $(p-1)!\equiv -1\mod p$
I can use Wilson's theorem in questions, and I can follow the proof whereby factors of $(p-1)!$ are paired up with their (mod $p$) inverses, but I am struggling to gain further insight into the meaning of the theorem.
To try and fi... | First let's understand why for prime $p$, $(p-1)!\equiv -1\mod p$. Understanding why that never happens for composite numbers actually ends up being a bit messy (although fundamentally not very complicated), so we'll cover that later.
The important intuition here is that the multiplicative group of $\mathbb Z/p\mathbb ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/478130",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
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Boolean Algebra-Simplification Assistance Needed I have to show that (!(P.Q) + R)(!Q + P.!R) => !Q by simplifying it using De Morgan's Laws. Here is what I did but I'm not sure it's right.
(!(P.Q) + R)(!Q + P.!R) => !Q
(!P + !Q + R)(!Q + P.!R)
!(P.Q) + !P.P.!R + !Q + !Q.P.!R + R.!Q + R.P.!R
!(P.Q) + 0 + !Q + !Q.P.!Q + ... | Indeed, your work is correct. Let's shorten things up, though, using the Distributive Law (D.L.) twice immediately following the application of DeMorgan's:
$\begin{align}(\overline{P\cdot Q} + R)\cdot (\overline Q + P\cdot \overline R) &= (\overline P + \overline Q + R)\cdot(\overline Q + P\cdot \overline R) \tag{DeMo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/478232",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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If $(ab)^3=a^3 b^3$, prove that the group $G$ is abelian. If in a group $G$, $(ab)^3=a^3 b^3$ for all $a,b\in G$, amd the $3$ does not divide $o(G)$, prove that $G$ is abelian.
I interpreted the fact that $3$ does not divide $o(G)$ as saying $(ab)^3\neq e$, where $e$ is the identity of the group.
As for proving $ab=ba... | This is an attempt at an elementary and linear exposition of this proof strategy, since the original seemed to cause some confusion from a comment - and it uses the language of homomorphisms, which I have avoided.
First you need to establish that every element of the group is a cube. Since the group has order not divis... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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$f(x,y)=\frac{x^3}{x^2+y^2}$ is not differentiable at $(0,0)$
Define $f(x,y)=\frac{x^3}{x^2+y^2}$ if $(x,y)\neq(0,0)$ and $f(x,y)=0$ for $(x,y)=(0,0)$
Show that it is not differentiable at $(0,0)$
I figured out that both $f_x$ and $f_y$ exists and are discontinuous at $(0,0)$ but can't say anything about different... | Were your function differentiable, it would be the case that if $f'(0;v)$ is the directional derivative at $0$ with direction $v$
$$f'(0;v+w)=f'(0;v)+f'(0,w)$$
Now let $v=(v_1,v_2)$ such that $v\neq 0$. Then
$$f'(0;v)=\lim\limits_{t\to 0}\frac{f(tv)}{t}=\lim_{t\to 0}\frac{t^3v_1^3}{t^3\lVert v\rVert ^2}=\frac{v_1^3}{\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/478368",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Finding minima and maxima of $\frac{e^{1/({1-x^2})}}{1+x^2}$
Find minima and maxima of $\frac{e^{1/({1-x^2})}}{1+x^2}$.
I have:
\begin{align}
f'(x)=\frac{ 2x\cdot e^{{1}/({1-x^2})} +\left(\frac{1+x^2}{(1-x^2)^2}-1\right)}{(1+x^2)^2}.
\end{align}
I have $x=0$ and $x=+\sqrt{3},x=-\sqrt{3}$ for solutions of $f'(x)=0$, ... | We calculate the derivative, because that is the first thing you did. It is fairly complicated. The denominator is $(1+x^2)^2$. The numerator, after some simplification, turns out to be
$$\frac{2x^3(3-x^2)}{(1-x^2)^2}e^{-1/(1-x^2)}.$$
This is indeed $0$ at $x=0$ and $x=\pm\sqrt{3}$.
Looking at the function: Note the sy... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/478467",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Two questions regarding $\mathrm {Li}$ from "Edwards" I would appreciate help understanding a relation in Edwards's "Riemann's Zeta Function."
On page 30 he has:
$$\int_{C^{+}} \frac{t^{\beta - 1}}{\log t}dt = \int_{0}^{x^{\beta}}\frac{du}{\log u}= \mathrm {Li} (x^{\beta}) - i\pi$$
He states for $\beta$ positive and r... | Here is a try for part two, the $- \pi i$ term:
Change variable from $u$ to $e^{t}$. Then $\log u = t$ and $du = e^{t}dt$.
Considering the integral around a half circle, now about and above $t = 0$, in the clockwise direction, the residue of the integrand is $-i\pi\left(e^{t}\right)_{z=0} = - i \pi$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/478534",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 2
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How many 3-digit positive integers are odd and do not contain digit 5? It is a GRE question. And it has been answered here. But I still want to ask it again, just to know why I am wrong.
The correct is 288.
My idea is, first I get the total number of 3-digit integers that do not contain 5, then divide it by 2. And beca... | (Hundreds) (Tens) (Units),
Units could be $(1, 3, 7, 9) \rightarrow 4$ numbers,
Tens could be $(0, 1, 2, 3, 4, 6, 7, 8, 9)\rightarrow 9$ numbers,
Hundreds could be $(1, 2, 3, 4, 6, 7, 8, 9) \rightarrow 8$ numbers,
(Hundreds) (Tens) (Units) $\rightarrow
(8) (9) (4) = 288$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/478619",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 4
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Find the shortest distance between the point $(8,3,2)$ and the line through the points $(1,2,1)$ and $(0,4,0)$ "Find the shortest distance between the point $(8,3,2)$ and the line through the points $(1,2,1)$ and $(0,4,0)$"
$$P = (1,2,1), Q = (0,4,0), A = (8,3,2)$$
$OP$ = vector to $P$
$$PQ_ = (0,4,0) - (1,2,1)$$
I fou... | Any point on the line looks like
$\vec{\rm r}\left(\lambda\right) \equiv \vec{P} + \lambda\vec{n}$ where
$\lambda \in {\mathbb R}$ and $\vec{n} \equiv \vec{Q} - \vec{P}$. The distance
between the point $\vec{A}$ and the point $\vec{\rm r}\left(\lambda\right)$ is given by
${\rm d}\left(\lambda\right)
=
\left\vert\vec{\r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/478765",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
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Example of an integrable function with an entire extension and whose derivative only vanishes at infinity I am looking for a function $f : \mathbb{C} \to \mathbb{C}$ with the following properties:
*
*$f$ is entire.
*$\int_{-\infty}^\infty |f(t)| \ dt < \infty$ i.e. the restriction of $f$ to the real line is in $L^1... | Let
$$
f(z)=\int_0^z\frac{\sin(w^2)}{w}\,dw-\frac{\pi}{4}.
$$
$f$ is an even entire function. As the real variable $t \to \infty$, $f(t)$ behaves like $O(t^{-2})$, so, in particular, that $\lim_{t\to\pm\infty}f(t)=0$ and $\int_{-\infty}^\infty|f(t)|\,dt<\infty$. On the other hand
$$
f'(z)=\frac{\sin(z^2)}{z}
$$
and
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/478816",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Generator for $[G,G]$ given that $G = \left$. If $S'$ is a generating set for $G$, let $S=S' \cup \{s^{-1} \,| \, s\in S\}$, then is the set $[S,S] = \{[s,z] \,|\, s,z \in S\}$ a generating set for the commutator subgroup $[G,G]$? I want to believe that this is true and "almost" have a proof, whose weak point might be ... | This won't be true in general. Suppose $G$ is generated by a set $S$ with only two elements. Then $[S,S]$ contains only two non-identity elements, inverse to one another, and so $\langle[S,S]\rangle$ will be cyclic.
But, for example, the symmetric group $S_n$ is generated by a set of two elements, but its commutator su... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/478887",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 1
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Show that an element has order $2$ in $S_n$ if and only if its cycle decomposition is a product of commuting $2$-cycles. In the Dummit-Foote text the algorithm for cycle decomposition is given as:
Based on the above algorithm the following exercise is asked to solve: Show that an element has order $2$ in $S_n$ if and ... | So all you need now is to show that any cycle can be written as the product of transposition...:
$$(i_1\,i_2\,\ldots\,i_n)=(i_2\,i_3)(i_3\,i_4)\cdot\ldots\cdot(i_{n-1}\,i_n)(i_n\,i_1)$$
Oberve that there are $\,n-1\,$ transpositions in the RHS above...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/478978",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Finding the local extrema of this trigonometric, multivariate function QUESTION
Find all extrema and their places for $$ f(x,y) = \mathtt{sin} x + \mathtt{cos} y + \mathtt{cos} (x-y)$$
for $ 0 \le x \le \frac{\pi}{2}$ and $ 0 \le y \le \frac{\pi}{2}$
ATTEMPT
I go ahead and find the first order partial derivatives:
$$... | You have found ($x = \pi/4$) values of $x$ where $\sin(x) = \cos(x)$. You need to find values of $x$ such that $\cos(x) = \sin(x/2)$ instead (i.e. the graphs of $\cos(x)$ and $\sin(x/2)$ intersect: http://www.wolframalpha.com/input/?i=plot+cos%28x%29%2C+sin%28x%2F2%29 ).
For instance, if $x = -\pi$, then $\sin(-\pi/2) ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/479164",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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De Moivre's formula I'm starting to study complex numbers, obviously we've work with De Moivre's formula. I was courious about the origin of it and i look for the original paper, I found it in the Philosophicis Transactionibus Num. 309, "De sectione Anguli", but only in latin, so some words are difficult to understand,... | DeMoivre only suggested the formula in $1722$. It was Euler who proved it in $1749$. Later DeMoivre's formula was discovered not only for complex numbers, but also for quaternions.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/479251",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Proving the following conditions are equivalent I have to prove four conditions are equivalent. I'm guessing I should proceed with (a) implies (b), (b) implies (c), (c) implies (d), and (d) implies (a)?
I have gotten (a) implies (b), (b) implies (a), and (b) implies (d). I'm struggling with (b) implies (c) and (c) impl... | $\bbox[5px,border:2px solid #4B0082]{(b)\implies (c)}$
$$\begin{align} a^{-1}b\in K&\implies (\exists k\in K)(a^{-1}b=k)\\
&\implies (\exists k\in K)(b=ak)\\
&\implies b\in aK \end{align}$$
$\bbox[5px,border:2px solid #4B0082]{(c)\implies (d)}$
Suppose $b\in aK$. There exists $k\in K$ such that $b=ak$.
*
*$\bbox[5px... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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A way to teach Archimedean property A student asked me how to understand the Archimedean property, I tried to re-read with him what he has already done in class (well, actually copy from the blackboard in class). However I think I'm not helping much, my suspicion is that maybe I'm being too formal. How can I approach t... | One way I have seen the Archimedean Property posed, which makes it relatively simple to understand, is as these two equivalent properties:
(1) For any positive number $c$, there is a natural number $n$ such that $n >c$.
(2) For any positive number $\epsilon$, there is a natural number $n$ such that $\frac{1}{n} < \epsi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/479401",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
} |
Function and Maclaurin series Function $f(x)=\frac{x^2+3\cdot\ e^x}{e^{2x}}$ need to be developed in Maclaurin series.
I can't find any rule to sum all fractions I've got...so any suggestion that helps?
Thanks
| Hint: Do you know the expansion for $e^{-2x}$ and $e^{-x}$? Can you multiply a power series by $x^2$ and by $3$? If so, you have the tools. Just give the series for $x^2e^{-2x}+3e^{-x}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/479522",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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is $\{(x,y) : x,y \in \mathbb{Z} \}$ a closed set? I claim yes, and to show this, it will suffice to show that $\mathbb{R}^2 \setminus \mathbb{Z}^2$ is open. So that for every $x \in \mathbb{R}^2 \setminus \mathbb{Z}^2$, we must find a neighborhood $N$ of $x$ such that $N \cap \mathbb{Z}^2 = \varnothing$. Let $r = \min... | $\mathbb{R}^2 \setminus \mathbb{Z}^2$ is a union of translated strips that look like $(0, 1) \times \mathbb R$ and $\mathbb R \times (0, 1)$, each of which is open.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/479597",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 1
} |
How does Cesaro summability imply the partial sums converge to the same sum? I can't reconcile this fact I used to know.
Suppose you have a sequence of nonnegative terms $a_k$. Let $s_n=\sum_{k=1}^n a_k$, and suppose
$$
\lim_{n\to\infty}\frac{s_1+\cdots+s_n}{n}=L.
$$
Then $\sum_{k=1}^\infty a_k$ also exists and equals ... | No this is not true. $a_k = (-1)^k$ is a counterexample.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/479654",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
what kind of singularity $e^{\sin z}$ has at $z=\infty$ Could anyone tell me what kind of singularity $e^{\sin z}$ has at $z=\infty$ ?
Enough to investigate $e^{\sin{1\over z}}$ at $z=0$
But $\lim_{z\to 0}$ the limit is $\infty$ and sometimes 0?
so essential singularity ?
| If it's an isolated singularity, and not removable, and not a pole, there's only one possibility left...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/479725",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Estimating the upper bound of prime count in the given range I need to estimate count of primes in the range $[n..m)$, where $n < m$, $n \in N$ and $m \in N$ and this estimation must always exceed the actual count of primes in the given range (i.e. be an upper bound).
The simple but ineffective solution would be just $... | One way to do it would be the following.
The exact number of primes below a given number $x$ is given by the prime counting function, denoted as $\pi(x)$. For example, $\pi(10) = 4$ and $\pi(20) = 8$. However, to find $\pi(x)$ it is required to count the number of primes explicitly, which is cumbersome.
Wikipedia infor... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/479798",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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} |
Show (via differentiation) $1-2+3-4+\cdots+(-1)^{n-1}n$ is $-\frac{n}{2}$ for $n$ even, $\frac{(n+1)}{2}$ for $n$ odd. i) By considering $(1+x+x^2+\cdots+x^n)(1-x)$ show that, if $x\neq 1$,
$$1+x+x^2+\cdots+x^n=\frac{(1-x^{n+1})}{1-x}$$
ii) By differentiating both sides and setting $x=-1$ show that
$$1-2+3-4+\cdots+(-1... | If $n$ is even, $n = 2k$ for some integer $k$. Then $(-1)^n = (-1)^{2k} = ((-1)^2)^k = 1^k = 1$ and $(-1)^{n + 2} = (-1)^n\times(-1)^2 = 1\times 1 = 1$. Therefore, we have
\begin{align*}
\frac{-2(n+1)(-1)^n + 1 +(-1)^{n+2}}{4} &= \frac{-2(n+1)\times 1 + 1 + 1}{4}\\
&= \frac{-2(n+1) + 2}{4}\\
&= \frac{-2n -2 + 2}{4}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/479878",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
In what manner are functions sets? From Introduction to Topology, Bert Mendelson, ed. 3, page 15:
A function may be viewed as a special case of what is called a relation.
Yet, a relation is a set
A relation $R$ on a set $E$ is a subset of $E\times E$.
while a function is a correspondence or rule. Is then a functio... | Let's begin at the other end.
A function can be regarded as a rule for assigning a unique value $f(x)$ to each $x$. Let's construct a set $F$ of all the ordered pairs $(x,f(x))$.
If $f:X\to Y$ we need every $x\in X$ to have a value $f(x)$. So for each $x$ there is an ordered pair $(x,y)$ in the set for some $y\in Y$. W... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/479936",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 1
} |
finding the limit $\lim\limits_{x \to \infty }(\frac{1}{e}(1+\frac{1}{x})^x)^x$ Can someone show me how to calculate the limit:
$$\lim_{x \to \infty }\left(\frac{1}{e}\left(1+\frac{1}{x}\right)^x\right)^x $$
I tried to use taylor series but failed.
Thanks
| Take the logarithm,
$$\begin{align}
\log \left(\frac{1}{e}\left(1+\frac{1}{x}\right)^x\right)^x &= x\left(\log \left(1+\frac1x\right)^x - 1\right)\\
&= x\left(x\log\left(1+\frac1x\right)-1\right)\\
&= x\left(-\frac{1}{2x} + O\left(\frac{1}{x^2}\right)\right).
\end{align}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/480003",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
Integral in spherical coordinates, $\Omega$ is the unit sphere, of $\iiint_\Omega 1/(2+z)^2dx\ dy\ dz$ $$\iiint_\Omega \frac{1}{(2+z)^2}dx\ dy\ dz$$
There is a VERY similar question How to integrate $\iiint\limits_\Omega \frac{1}{(1+z)^2} \, dx \, dy \, dz$ here
But this is different.
I like my spherical coordinates to... | $$I=\int^1_0\int^\frac{\pi}{2}_{-\frac{\pi}{2}}\frac{\cos(\theta)p^2}{(2+p\sin(\theta))^2}\int^{2\pi}_0d\psi\ d\theta\ dp$$
$$=2\pi\int^1_0p\int^\frac{\pi}{2}_{-\frac{\pi}{2}}\frac{p\cos(\theta)}{(2+p\sin(\theta))^2}d\theta\ dp$$
Let $u=2+p\sin(\theta)$ then $\frac{du}{d\theta}=p\cos(\theta)$
Thus: $$du=p\cos(\theta)d\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/480129",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Gradiant of XAY with respect to A How can I find the gradient of the following function with respect to A?
$$
F(A) = X^T \cdot A \cdot Y
$$
Where X and Y are mx1 vectors and A is mxm matrix
| Writing
$F(A + \Delta A) = X^T A Y + X^T \Delta A Y$,
we see that
$F(A + \Delta A) - F(A) = X^T \Delta A Y$;
the error terms vanish exactly as they would for a plain old vanilla-flavored scalar function $f(x) = ax$:
$f( x + \Delta x) - f(x) = a(x + \Delta x) - ax = a \Delta x$:
thus the derivative is a constant linear... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/480207",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Any finite set is a null-set How can we prove that a finite set is a null-set? Maybe would it be easier to prove that the outer measure of a finite set is $0$? any ideas on how to tackle this problem?
thanks,
| You can just as easily prove that a countably infinite set $\{a_1,a_2,a_3,\dots\}$ is null by putting an interval of width $\frac{\epsilon}{2^n}$ about $a_n$.
The above paragraph assumed we were working in the reals, but a similar idea works for $\mathbb{R}^n$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/480256",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Predicate Calculus statement I've been asked to write down a statement using predicate calculus and it is confusing me a great deal.
I've got statement A "no dog can fly" and B "There is a dog which can fly"
D = set of all dogs , F = set of all creatures that can fly
P(x) is the proposition that "creature x can fly"
Q... | The statement A is OK, apart from a missing parenthesis. Statement B should be something like $\exists x(Q(x)\land P(x))$.
Your version of B would be true if there were, for example, no flying creatures.
There are always many equivalent ways of stating things. Closer in tone to the English statement of A is $\forall x(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/480332",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Two functions agreeing except on set of measure zero
Let $f,g:S\rightarrow\mathbb{R}$; assume $f$ and $g$ are integrable over $S$. Show that if $f$ and $g$ agree except on a set of measure zero, then $\int_Sf=\int_Sg$.
Since $f$ and $g$ are integrable over $S$, we have $f-g$ also integrable over $S$. So $(f-g)(x)=0$ ... | If you want to go with all the details starting from the definition, do it like this :
$$
\left| \int_S (f - g) \, d\mu \right| \le \int_S |f-g| d\mu = \sup \left\{ \left. \int_S \varphi \, d\mu \, \right| \, 0 \le \varphi \le |f-g|, \varphi \in \mathcal L(S)\right\}
$$
where I wrote $\mathcal L(S)$ for the set of al... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/480403",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 0
} |
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