Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Show that a positive operator on a complex Hilbert space is self-adjoint Let $(\mathcal{H}, (\cdot, \cdot))$ be a complex Hilbert space, and $A : \mathcal{H} \to \mathcal{H}$ a positive, bounded operator ($A$ being positive means $(Ax,x) \ge 0$ for all $x \in \mathcal{H}$).
Prove that $A$ is self-adjoint. That is, pro... | You should apply the polarization identity in the form
$$4(Ax,y) = (A(x+y),x+y) - (A(x-y),x-y) -i(A(x+iy),x+iy) + i(A(x-iy),x-iy).$$
Since you already know $(Az,z) = (z,Az)$ for all $z \in \mathcal{H}$, it is not difficult to deduce $A^\ast = A$ from that.
| {
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"timestamp": "2023-03-29T00:00:00",
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Uniform convergence of a family of functions on $(0,1)$ Let the family of functions be
$$f_n(x) = \dfrac{x}{1+nx}.$$
Is the sequence $f_n$ uniformly convergent in the interval $(0,1)$?
| $\frac{x}{1 + nx} = \frac{1}{\frac{1}{x} + n} \leq \frac{1}{n}$ which doesn't depend on $x$ hence your sequence converges uniformly to $0$
| {
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Mistake in simplification of large polynomial inequality? We are to solve for $p$, and the inequality to simplify is
$$10p^3(1-p)^2 + 5p^4(1-p) + p^5 - 3p^2(1-p) - p^3 > 0$$
On the next line of the textbook, the author simplifies this expression to
$$3(p-1)^2(2p-1) > 0 \implies p > \frac{1}{2}$$
Since no work was sho... | The original expression factorizes as $4p^2(p-1)(p-3)(p-\frac14)>0.$ So $\frac14<p<1$ or $p>3.$
| {
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Mean and Var of a gamma distribution Let X have a Gamma distribution with a known scale
parameter 1, but an unknown shape parameter, that itself is random,
and has the standard exponential distribution.
How do I compute the mean and the variance of X?
Thanks!
| If we call the unknown parameter $\theta$, then what you are seeking is
$$
E(X)=\int_0^\infty x f(x)dx=\int_0^\infty x \int_0^\infty f(x, \theta)d\theta dx=\int_0^\infty x\int_0^\infty f(x|\theta)f(\theta)d\theta dx
$$
where then it is given that
$$
\begin{align*}
x|\theta&\sim Gamma(1, \theta)\\
\theta&\sim Exp(1)
\e... | {
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Show $\exp(A)=\cos(\sqrt{\det(A)})I+\frac{\sin(\sqrt{\det(A)})}{\sqrt{\det(A)}}A,A\in M(2,\mathbb{C})$
Show $$\exp(A)=\cos(\sqrt{\det(A)})I+\frac{\sin(\sqrt{\det(A)})}{\sqrt{\det(A)}}A$$
for $A\in M(2,\mathbb{C})$. In addition, $\operatorname{trace}(A)=0$.
Can anyone give me a hint how this can connect with cosine ... | The caracteristic polynomial of $2\times 2$ matrix $A$ is
$$X^2-\mathrm{Tr}(A)X+\mathrm{det}(A),$$
so that a trace $0$ matrix satisfies the equation
$$A^2=-\mathrm{det}(A)I_2.$$
Let $\lambda\in\Bbb C$ be a square root of $\det(A)$. It follows from the equation above that for every integer $p$
$$A^{2p}=(-1)^p\lambda^{2... | {
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Prove or disprove: if $A$ is nonzero $2 \times 2$ matrix such that $A^2+A=0$, then A is invertible
if $A$ is nonzero $2 \times 2$ matrix such that $A^2+A=0$, then A is invertible
I really can't figure it out. I know it's true but don't know how to prove it
| This is not true. For example
$$
A=\begin{pmatrix}
-1&0\\
0&0
\end{pmatrix}
$$
| {
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How prove this inequality $\sum\limits_{k=1}^{n}\frac{1}{k!}-\frac{3}{2n}<\left(1+\frac{1}{n}\right)^n<\sum\limits_{k=1}^{n}\frac{1}{k!}$ show that
$$\sum_{k=1}^{n}\dfrac{1}{k!}-\dfrac{3}{2n}<\left(1+\dfrac{1}{n}\right)^n<\sum_{k=0}^{n}\dfrac{1}{k!}(n\ge 3)$$
Mu try: I konw
$$\sum_{k=0}^{\infty}\dfrac{1}{k!}=e$$
and I ... | I think you mean this inequality$$\sum_{k=0}^{n}\dfrac{1}{k!}-\dfrac{3}{2n}<\left(1+\dfrac{1}{n}\right)^n.$$In fact, we can prove a sharper one $$\left(1+\frac{1}{n}\right)^n+\frac{3}{2n}>e.$$Let $f(x)=\left(1+\dfrac{1}{x}\right)^x+\dfrac{3}{2x}$, then$$f'(x)=\left(1+\dfrac{1}{x}\right)^x\left(\ln\left(1+\frac{1}{x}\ri... | {
"language": "en",
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Largest triangle to fit in a circle will be isosceles triangle? Largest triangle to fit in a circle will be isosceles triangle?
Or some other type?
| Yes, what you say is true, but you can say more than that.
Given a particular chord of a circle, you can maximize the area of the triangle by having the third vertex as far away as possible (area is half base times perpendicular height), which means that it will be on the perpendicular bisector of the chord where it cr... | {
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Reflexive, separable containing all finite dimensional spaces almost isometrically Is there a separable, reflexive Banach space $Z$ such that for every finite dimensional space $X$ and every $a>0$, there is a $1+a$-embedding of $X$ into $Z$?
I can do the question without the 'reflexive' (in which case it's true), but I... | I thought I would mention a different answer to Norbert's since the paper containing the result I cite is not that well known and deserves to be advertised. Szankowski has shown that there exists a sequence of Banach spaces $X_m$, $m\in\mathbb{N}$, each isomorphic to $\ell_2$, with the following property: every finite ... | {
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Which is larger? $20!$ or $2^{40}$? Someone asked me this question, and it bothers the hell out of me that I can't prove either way.
I've sort of come to the conclusion that 20! must be larger, because it has 36 prime factors, some of which are significantly larger than 2, whereas $2^{40}$ has only factors of 2.
Is the... | You can also separate each one of these, into 10 terms and note:
$$1\times 20 > 2^4$$
$$2\times 19 > 2^4$$
$$\vdots$$
$$10 \times 11 >2^4$$
$$\Rightarrow 20! > 2^{40}$$
The idea is to break the factorial symmetrically into smaller pieces; which is not the most robust method for inequalities which include factorials; bu... | {
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Swap two integers in different ways This is a famous rudimentary problem : how to use mathematical operations (not any other temporary variable or storage) to swap two integers A and B. The most well-known way is the following:
A = A + B
B = A - B
A = A - B
What are some of the alternative set of operations to achieve... | I'm not sure if you're asking for all solutions or not, but one of the most famous solutions is by using binary xor three times. $A=A\oplus B,B=A\oplus B,A=A\oplus B$.
| {
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What is the value of sin(arcsin(4))? In this case arcsin() is the unrestricted sin inverse function. I know that it is either undefined or has the value of 4. It could be undefined because arcsin() has only a doman of -1...1 and 4 is out of the domain. On the other hand, it could be that since they are inverses the int... | Complex values aside, this expression cannot be evaluated since the $arcsin$ can only be taken from valus between -1 and 1 (inclusive) so the cancelation property of the inverses cannot be applied here. Cancelation property of inverses can be used for values that are in the respective domains of the functions. Think ab... | {
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Verification of the existence of an inverse in a group In a group $(G,*)$ with neutral element $e$, does the relation $x*y=e$ imply $y*x=e$? i think it is true. Indeed, $y*x=y*(x*y)*x=(y*x)*(y*x)$ hence $(y*x)^n=y*x$ for each $n\geq 2$ which is true only if $y*x=e$. Is this correct? If yes, then to verify the existence... | I'd say that the step where you go from $(y * x)^n = y * x$ for each $n \geq 2$ to $y * x = e$ needs some reasoning.
However, from $y * x = (y * x) * (y * x)$ you can immediately get $y * x = e$ by the following proposition.
Proposition. Let $g$ be an idempotent element of some group $G$, i.e., $gg = g$. Then $g = e$.
... | {
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Given a theorem can it always be reduced logically to the axioms? It's probably a silly question but I’ve been carrying this one since infancy so i might as well ask it already.
let ($p \implies q$) be a theorem where $p$ is the hypotheses and $q$ is the conclusion. If stated in logical symbols can it always be reduce... | This question is quite natural, as the notion of "rigorous proof" largely depends on the context in which it is mentioned.
The short answer is Yes. Every mathematical proof, if correct, can be formulated as a derivation starting from axioms (usually of ZFC), and using basic deduction rules.
In practice, it is of cour... | {
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Do I influence myself more than my neighbors? Consider relations between people is defined by a weighted symmetric undirected graph $W$, and $w_{ij}$ shows amount of weight $i$ has for $j$. Assume all weights are non-negative and less than $1$ i.e. $$0\leq w_{ij}<1, \forall{i,j}$$ and symmetric $w_{ij}=w_{ji}$. We say ... | Look at the inverse of $$\left[\begin{array}{ccc}1 & 0 & .9\\ 0 & 1 & .9\\ 0 & .9 & 1\end{array}\right]$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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consecutive convergents Problem: Let $\phi=\frac{1+\sqrt{5}}{2}$ be the golden ratio and let $a$, $b$, $c$, $d$ be positive integers so that $\frac{a}{b}>\phi>\frac{c}{d}$. It is also known that $ad-bc=1$. Prove that $a/b$ and $c/d$ are consecutive convergents of $\phi$.
Numerical experimentations point towards the va... | Note that $\phi=1+\frac{1}{1+\frac{1}{1+\ldots}}$ has convergents $\frac{f_{n+1}}{f_n}$, i.e. ratios of consecutive Fibonacci numbers. Note that I have used lower case for the Fibonacci numbers so as to avoid confusion with the Farey sequence $F_n$.
The main idea is to appeal to the properties of Farey sequences.
*
... | {
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If $[K : \mathbb{Q}] = 2$, then $K = \mathbb{Q}(\sqrt{d})$ This isn't for homework, but I would just a like a hint please. The question asks
If $K$ is an extension field of $\mathbb{Q}$ and $[K : \mathbb{Q}] = 2$ (the dimension of $K$ over $\mathbb{Q}$), then $K = \mathbb{Q}(\sqrt{d})$ for some square-free integer $d... | $v$ satisfies a quadratic equation $ax^2 + bx + c = 0$, where $a \ne 0, b, c \in \mathbb{Z}$.
Solve this equation and deduce $K = \mathbb{Q}(\sqrt D)$, where $D = b^2 -4ac$.
| {
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what is the relationship between vector spaces and rings? Can you show me an example to show how vector and scalar multiplication works with rings would be really helpful.
| The relationship between a ring and its modules is the analogue of the relationship of a field and its vector spaces.
For a field (or even skewfield) $F$, the Cartesian product $F\times F\times \dots\times F$ of finitely many copies of $F$ is a vector space in the ways you are probably familiar with.
There is no reason... | {
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Proving $\lim _{x \to 0^-} \frac{1}{x}=-\infty$ How do I calculate the limit:
$$\lim _{x \to 0^-} \frac{1}{x}$$
The answer is clearly $-\infty$, but how do I prove it? Its clear that as x approaches 0 from the right side, $x$, becomes infinitely small, but negative, making the limit go to $-\infty$, but how do I prove ... | show x = -.0001 then x = -.00000001 then x = -.00000000000000001. Choose values that get closer and closer to 0 then graph that. That is the best way I could show it. Is this what you mean? Or am I making this too simple?
| {
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Trying to understand the equivalence of two definitions of a sieve. Let $\mathcal{C}$ be a small category, let $C$ be an object of $\mathcal{C}$ and let $\mathbf{y}:\mathcal{C}\to[\mathcal{C}^{op},\mathbf{Set}]$ be the Yoneda embedding.
I am trying to derive the simple fact that a sieve $S$ on $C$ is a family of morph... | The monic $S \rightarrow y(C)$ determines subsets $S(A) \subseteq Hom_\mathcal{C}(A,C)$, in the standard sense (why?), so that composing with a morphism sends the subset $S(A)$ to $S(B)$. Since the category is small, you can also take the union $\bigcup_{A \in \mathcal{C}} S(A)$. You can then check that this union is p... | {
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On $n\times n$ matrices $A$ with trace of the powers equal to $0$
Let $R$ be a commutative ring with identity and let $A \in M_n(R)$ be such that $$\mbox{tr}A = \mbox{tr}A^2 = \cdots = \mbox{tr}A^n = 0 .$$ I want to show that $n!A^n= 0$.
Any suggestion or reference would be helpful.
P.S.: When $R$ is a field of ch... | The following argument also works in prime characteristic. The coefficients
of the characteristic polynomial
$$
\chi(t)=\sum^n_{j=0}
(-1)^j \omega_j (A)\:t^{n-j}\;
$$ of $A$ satisfy the following identities:
$$
\sum^j_{i=1} (-1)^{i+1} {\rm tr}(A^i)\:\omega_{j-i} (A) =j\cdot \omega_j (A)
\hbox{ with } \omega_0 (A)=1,... | {
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Find a non-principal ideal (if one exists) in $\mathbb Z[x]$ and $\mathbb Q[x,y]$ I know that $\mathbb Z$ is not a field so this doesn't rule out non-principal ideals. I don't know how to find them though besides with guessing, which could take forever.
As for $\mathbb Q[x,y]$ I know $\mathbb Q$ is a field which would ... | Here is a general result:
If $D$ is a domain, then $D[X]$ is a PID iff $D$ is a field.
One direction is a classic result. For the other direction, take $a\in D$, consider the ideal $(a,X)$, and prove that it is principal iff $a$ is a unit.
This immediately answers both questions: $(2,X)$ is not principal in $\mathbb ... | {
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Intuition/Real-life Examples: Pairwise Independence vs (Mutual) Independence Would someone please advance/discuss some real-life situations falsities $1, 2$?
I'd like to intuit why these are false. As a neophyte, since I still need to compute the probabilities for the examples in the two answers to apprehend them, I ha... | Head appears in the first toss.
B: Head appears in the second toss.
C: Head appears in the third toss.
D: A and B yield the same outcome.
Mutual independence:
Firstly we only consider the $A, B, C$ (just treat $D$ as nonexistent). It is obvious that they are mutual independent. And here are two perspectives of this sta... | {
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Solve $\frac{d}{dx}f(x)=f(x-1)$ I am trying to find a function such that
$\dfrac{d}{dx}f(x)=f(x-1)$
Is there such function other than $0$ ?
| Certainly. Let $b$ be the unique real number such that $b=e^{-b}.$ Then for any real $a,$ the function $$f(x)=ae^{bx}$$ satisfies the desired property. In fact, for real-valued functions on the reals, only functions of this form will satisfy the desired property. (As achillehui points out in the comments, there are oth... | {
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$\sum_{i=1}^n \frac{1}{i(i+1)} = \frac{3}{4} - \frac{2n+3}{2(n+1)(n+2)}$ by induction. I am wondering if I can get some help with this question. I feel like this is false, as I have tried many ways even to get the base case working (for induction) and I can't seem to get it. Can anyone confirm that this is false? If ... | Let $$F(n)=\sum_{i=1}^n\frac{1}{i(i+1)}$$
and $$G(n)=\frac{3}{4}-\frac{2n+3}{2(n+1)(n+2)}$$
Your task is to prove that $F(n)=G(n)$ for all $n$. To do this by induction, prove first that $F(1)=G(1)$. Then, assume $F(n)=G(n)$. Add $\frac{1}{(n+1)(n+2)}$ to both sides; this is because $F(n)+\frac{1}{(n+1)(n+2)}=F(n+1)$... | {
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Each closed set is $f^{-1}(0)$ Let $X$ be some compact Hausdorff space (or even normal space). Is it true that each closed subset $X'$ is $f^{-1}(0)$ for some $f\in C(X,\mathbb{R})$? I know that there is Urysohn's lemma which gives us an opportunity to continue each function $X'\longrightarrow \mathbb{R}$ to a function... | No. The compact ordinal space $\omega_1+1$ with the order topology is an easy counterexmple, as is $\beta\Bbb N$: each has at least one point $x$ such that $\{x\}$ is not a $G_\delta$-set and therefore cannot be $f^{-1}[\{0\}]$ for any continuous real-valued $f$.
You’re asking for spaces in which every closed set is a... | {
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Expectation value of certain number of trials of multinomial distribution. Player can extract card from deck (the size of deck is infinite) to obtain one of $k$ kinds of cards, and the possibility of obtaining each kind is given by $p_i$. (Obviously $\sum_{i=1}^{k} p_{i} = 1$). If you collect AT LEAST one card from all... | Please search the web for the coupon collector's problem.
Let $ N$
be the number of cards to extract till we get all $ k$
kinds, and let $N_{i}$
be the number of extracts to collect the $i'th$
kind after $i−1$
kinds have been collected - note that $N=\sum_{i=1}^{k}N_{i}.$
We want to get more then m
dollars ... | {
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Prove that the Gale-Shapley algorithm terminates after at most $n^2 - n + 1$ proposals. How do you prove that the Gale-Shapley algorithm terminates after at most $n^2 - n + 1$ proposals by showing that at most one proposer receives his or her lowest-ranked choice?
| Assuming you are using the same number of proposers and acceptors (because all of your problems are this way):
If exactly one proposer (from now on man) gets his last choice woman, he will have proposed $n$ times. The remaining $n-1$ men are able to propose a maximum of $n-1$ times so $$(n-1)(n-1)+n=n^2-2n+1+n=n^2-n+1... | {
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Quadratic residues solutions I have a homework question that I can't figure out. It says:
If the prime p > 5 , show that there are always two quadratic residues of p that differ by two
| Note that $1$ and $4$ are QR. So if one of $2$ or $3$ is a QR, we are finished.
If both $2$ and $3$ are NR, then $6$ is a QR. But then $4$ and $6$ are QR that differ by $2$.
| {
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A positive integer is divisible by $3$ iff 3 divides the sum of its digits I am having trouble proving the two following questions:
*
*If $p|N$, $q|N$ and gcd(p,q)=1, then prove that $pq|N$
*If $x$ is non zero positive integer number, then prove that $3|x$ if and only if 3 divides the sum of all digits of $x$.
For... | Here is an explanation of number 2. We will use a corollary from a theorem (both taken from Rosen's Discrete Mathematics text.
Theorem: Let $m$ be a positive integer. If $a\equiv b \text{ mod } m$ and $c\equiv d \text{ mod } m,$ then
$$
a+c \equiv b+d \text{ mod } m \qquad\text{and}\qquad ac \equiv bd \text{ mod } m... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$x^a+y^b$ is divisible by $x^b+y^a$ for infinitely many $(x,y)$ Let $a\geq b>0$ be integers. For which $(a,b)$ do there exist infinitely many positive integers $x\neq y$ such that $x^a+y^b$ is divisible by $x^b+y^a$?
If $a=b$, we certainly have $x^a+y^a$ divisible by itself.
For $a>b$ maybe we can choose some form of $... | If $a$ is odd and $b=1$ and any positive $x>1$ with $y=1$ you have the integer $(x^a+1)/(x+1).$ There are other families of pairs $(a,b)$ with some good choices of $x$ or $y$. For example replacing $x$ above by $x^t$ gives the pair $(at,t)$ of exponents which, on choosing $y=1$, gives the integer $(x^{at}+1)/(x^t+1).$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Complex Analysis Proofs Let $f$ = $u + iv$ be an entire function satisfying $u(x,y) \geq v(x,y)-1$ for every $(x,y) \in R^2$. Prove that all functions $f, u, v$ are constant.
Can someone please help me prove this...
| With these type of questions, when I see entire and some bound on the function, I immediately try to apply Liouville's theorem. From there, it is just a matter of trying to get a bounded entire function. In this case, the following gives a bounded entire function:
The condition that $u\geq v-1$ can be rewritten to say ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Question about Ito integral I was wondering if Ito integral:
$\int_0^T B(t)dB(t) $
is Gaussian (in which B(t) is Brownian Motion)??
Thank you so much, I appreciate any help ^^
| $d(B_t^2) = 2B_t dB_t + dt$. Therefore your integral is $\frac12(B_T^2-T)$. $B_T$ is Gaussian $N(0,\sqrt T)$, therefore $B_T^2$ is $T$ times a $\chi^2_1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/564942",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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The Concept of Instantaneous Velocity The concept of instantaneous velocity really becomes counter-intuitive to me when I really think deeply about it. Instantaneous velocity is the velocity of something at an instant of time; however, at the very next instant the velocity changes. In general, speed tells us how quickl... | Anything that changes in speed will do so as a result of momentum transfer into the object. This requires some interaction with other forces. Force fields are the source of momentum transfer, this transfer takes some time, thus we get acceleration.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/565003",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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Convergence of $\frac{1}{n}\sum_{i=1}^{n}\left[\frac{\left(\log\left(1+i/n\right)\right)^2}{1+i/n}\right]$ Sequence of real numbers
$$S_n=\frac{1}{n}\sum_{i=1}^{n}\left[\frac{\left(\log\left(1+\frac{i}{n}\right)\right)^2}{1+\frac{i}{n}}\right]$$
Does $\lim\limits_{n \to \infty} S_n$ exist? If so, compute the value.
My ... | the B) methods is true.
$$\lim_{n\to\infty}\dfrac{1}{n}\sum_{i=1}^{n}\left(\dfrac{(\ln{1+\dfrac{i}{n}})^2}{1+\dfrac{i}{n}}\right)=\lim_{n\to\infty}\dfrac{1}{n}\sum_{i=1}^{n}f\left(\dfrac{i}{n}\right)=\int_{0}^{1}f(x)dx$$
where $$f(x)=\dfrac{\left(\ln{(1+x)}\right)^2}{1+x}$$
| {
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"timestamp": "2023-03-29T00:00:00",
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Does $\lim_{(x,y) \to (0,0)} xy^4 / (x^2 + y^8)$ exist? From this question on answers.yahoo, the guy says the following limit does not exist:
$$\lim_{(x,y) \to (0,0)} \frac{xy^4}{x^2 + y^8},$$
then on wolfram, it says the limit is equal to $0$. When I did it myself, I tried approaching $(0,0)$ from the $x$-axis, $y$-a... | Hint: let $x=ky^4(k\neq 0)$,so
$$\lim_{(x,y)\to(0,0)}\dfrac{xy^4}{x^2+y^8}=\dfrac{k}{k^2+1}$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Let p and q be distinct odd primes. Define $n=pq$ and$ \phi(n)=(p−1)(q−1)$ (a) Show that
$p+q = n−\phi(n)+1$ and $p−q = \sqrt{(p+q)^2−4n}$.
(b) Suppose you are given that $n = 675683$ and are told that $p−q = 2$. Explain how this information can help us factor $n$ quickly.
(Hint: Try to use the result from part (a)... | Since you already have expressions for p+q and p-q in terms of just n and phi(n), all you have to do is use these expressions and note that p can be expressed as
p=((p+q)+(p-q))/2
and likewise
q=((p+q)-(p-q))/2
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/565273",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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does $(\overline{E})^{'}= E^{'} \cup ( E^{'})^{'}$ holds? My question is as follows:
Suppose $E$ is a set in metric space $X$, let $\overline{E}$ denote the closure of E, let $E^{'}$ be the set of all the limit points of $E$. We all know that $\overline{E}=E\cup E^{'} $ Then my question is: Does the following equality... | We have more. In a Hausdorff space (even in a $T_1$ space, if you already know what that is), $x$ is a limit point of $A$ if and only if every neighbourhood of $x$ contains infinitely many points of $A$. Thus in such spaces, we have
$$(\overline{E})' = E'.$$
Since evidently $(E')' \subset (\overline{E})'$, the equality... | {
"language": "en",
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Intuitive understanding of determinants?
For a $n \times n$ matrix $A$:$$\det (A) = \sum^{n}_{i=1}a_{1i}C_{1i}$$
where $C$ is the cofactor of $a_{1i}$. If the determinant is $0$, the matrix is not invertible.
Could someone an intuitive explanation of why a zero determinant means non-invertibility? I'm not looking for... | There is a simple property for determinants:
$Det(A)Det(b)=Det(AB)$, so if you take:
$Det(A A^{-1})= Det(A)Det(A^{-1})= Det(I) = 1$, from this follows than $det(A)$ must be different from $0$ to be invertible.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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What's the limit of $(1+\frac{1}{8^n})^n$ What's the limit of $(1+\frac{1}{8^n})^n$? How do I find the answer for this?
Thanks in advance.
| $\newcommand{\+}{^{\dagger}}%
\newcommand{\angles}[1]{\left\langle #1 \right\rangle}%
\newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}%
\newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}%
\newcommand{\dd}{{\rm d}}%
\newcommand{\isdiv}{\,\left.\right\vert\,}%
\newcommand{\ds}[1]{\displaystyle{#1}}%
\new... | {
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Algebraic solution to: Do the functions $y=\frac{1}{x}$ and $y=x^3$ ever have the same slope? The exercise doesn't specify how it must be answered, so I chose a graphical proof because I couldn't come up with an algebraic one. Sketching the graphs of $y=\frac{1}{x}$ and $y=x^3$, I noticed that $y=x^3$ always has a nonn... | Let $f(x)=x^3$ and $g(x)=x^{-1}$. We have $f'(x)=3x^2$ and $g'(x)=-x^{-2}$, now we do exactly what you said:
$Im(f')\cap Im(g')=\left\{t\in\mathbb{R}|t\geq0 \right\}\cap\left\{t\in\mathbb{R}|t<0 \right\}=\left\{t\in\mathbb{R}|t\geq0\wedge t<0 \right\}=\emptyset$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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PDF of $Y - (X - 1)^2$ for $(X, Y)$ uniform on $[0, 2] \times [0, 1]$ I am trying to find the p.d.f (but will calculate the c.d.f first) of $Z = Y - {(X - 1)}^2$ knowing that $(X, Y)$ is distributed uniformly on $[0, 2] \times [0, 1]$. So,
$$f_{X, Y}(x, y) = \begin{cases}\frac{1}{2} & (x, y) \in [0, 2] \times [0, 1] \\... | Given: the joint pdf of $(X,Y)$ is $f(x,y)$:
(source: tri.org.au)
Some neat solutions have been posted showing all the manual steps which involve some work. Alternatively (or just to check your work), most of this can be automated which makes solving it really quite easy. Basically, it is a one-liner:
The cdf of $Z =... | {
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"source": "stackexchange",
"question_score": "5",
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Accumulation Points for $S = \{(-1)^n + \frac1n \mid n \in \mathbb{N}\}$ I was recently asked to find the accumulation points of the set $$S = \{(-1)^n + \frac{1}{n} \mid n \in \mathbb{N}\}$$
I answered that the accumulation points are $\{-1,1\}$, because despite the fact that $\frac{1}{n}$ diverges, we can still use $... | Let's denote
$$a_n=(-1)^n+\frac{1}{n}$$
then the subsequence $(a_{2n})$ and $(a_{2n+1})$ are convergent to $1$ and $-1$ respectively then $1$ and $-1$ are two accumulation points. There's not other accumulation point since any convergent subsequence of $((-1)^n)$ has either $1$ or $-1$ as limit.
| {
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"timestamp": "2023-03-29T00:00:00",
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Elementary proof that $*$homomorphisms between C*-Algebras are norm-decreasing A lecturer once gave a very elementary proof that $*$-homomorphisms between C*-algebras are always norm-decreasing. It is well-known that this holds for a $*$-homomorphism between a Banach algebra and a C*-algebra, but all the proofs I find ... | Let $f\colon A\to B$ be a *-homomorphism. Let us note that $f$ cannot enlarge spectra of self-adjoint elements in $A$, that is, for all $y\in A$ self-adjoint we have $\mbox{sp}(f(y))\setminus \{0\} \subseteq \mbox{sp}(y)\setminus \{0\}$. By the spectral radius formula, we have $\|y\|=r(y)$. Now, let $x\in A$. It follow... | {
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For what values of $r$ does $y=e^{rx}$ satisfy $y'' + 5y' - 6y = 0$? For what values of $r$ does $y=e^{rx}$ satisfy $y'' + 5y' - 6y = 0$?
Attempt:
$y' = [e^{rx}] (r)$
$y''= r^2e^{rx}$
| If you plug them in, you obtain : $$r^2+5r-6=0$$
Solving this equation you get $r=1$ or $r=-6$.
That means that the general solution of the suggested ODE is : $$y(x)=ae^t + be^{-6t}, (a,b) \in \Bbb R^2$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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If $Y$ and $Y \cup X$ are connected. Must their be some $X$-component ($C$) where $C \cup Y$ is connected? This is a question I had, while trying to solve a homework problem. My original approach was dependent upon the following statement being true.
If $Y$ and $Y \cup X$ are connected, then there is some connected com... | There certainly is some connected component of $X$ where $Y \cup C$ is connected. If there wasn't, then you would get a disjoint union of $X$ and $Y$, a contradiction..
Proof. Choose an arbitrary $x$ in $C$ with the restriction it is the closest point to $Y$ (for some point $y \in Y$). For it to be connected there woul... | {
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If $\gcd(a, b) = 1$ then $\gcd(ab, a+b) = 1$? In a mathematical demonstration, i saw:
If $\gcd(a, b) = 1$ Then $\gcd(ab, a+b) = 1$
I could not found a counter example, but i could not found a way to prove it too either.
Could you help me on this one ?
| First prove
*
*$\gcd(mn, k)=1$ if and only if $\gcd(m,k)=1$ and $\gcd(n,k)=1$.
*If $\gcd(m,k)=1$ then $\gcd(m,m+k)=1$.
The desired result follows from these like so: From $\gcd(a,b)=1$, we have $\gcd(a, a+b)=\gcd(b, a+b)=1$, implying
$\gcd(ab, a+b)=1$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Infinite regular sets Would it be true that for all infinite regular subsets, each one contains subsets that are not c.e/r.e (countably enumerable/recursively enumerable)?
Intuitively this seems true because of sets that are uncountable.
| Uncountability has nothing to do with it: none of the sets that you’re talking about in this context is uncountable. However, the statement is true; it follows from the pumping lemma for regular languages. If $L$ is regular and infinite, let $p$ be its pumping length, and let $w\in L$ be such that $|w|\ge p$. Then $w$ ... | {
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Compute $\int_0^{\infty}\frac{\cos(\pi t/2)}{1-t^2}dt$ Compute $$\int_0^{\infty}\frac{\cos(\pi t/2)}{1-t^2}dt$$
The answer is $\pi/2$. The discontinuities at $\pm1$ are removable since the limit exists at those points.
| Define
$$f(z)=\frac{e^{\pi iz/2}}{1-z^2}\;,\;\;C_R:=[-R,-1-\epsilon]\cup\gamma_{-1,\epsilon}\cup[1+\epsilon,\epsilon]\cup\gamma_{1\epsilon}\cup[1+\epsilon,R]\cup\Gamma_R$$
with $\;\epsilon, R\in\Bbb R^+\;$ and
$$\gamma_{r,s}:=\{r+se^{it}\;;\;0\le t\le \pi\}\;,\;r,s\in\Bbb R^+\;,\;\Gamma_R:=\{Re^{it}\;;\;0\le t\le \pi\}... | {
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Closure of equivalence relations Show that the transitive closure of the symmetric closure
of the reflexive closure of a relation R is the smallest
equivalence relation that contains R.
I can understand the statement intuitively but can't come up with a mathematical proof
| HINT: Let $S$ be the transitive closure of the symmetric closure of the reflexive closure of $R$. You have to show three things:
*
*$R\subseteq S$.
*$S$ is an equivalence relation.
*If $E$ is an equivalence relation containing $R$, then $E\supseteq S$.
The first of these is pretty trivial, and the second isn... | {
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What is $\int_0^1\frac{x^7-1}{\log(x)}\mathrm dx$? /A problem from the 2012 MIT Integration Bee is
$$
\int_0^1\frac{x^7-1}{\log(x)}\mathrm dx
$$
The answer is $\log(8)$. Wolfram Alpha gives an indefinite form in terms of the logarithmic integral function, but times out doing the computation. Is there a way to do it by ... | Yet another direct way forward is to use Frullani's Integral. To that end, let $I(a)$ be the integral given by
$$I(a)=\int_0^1 \frac{x^a-1}{\log x}\,dx$$
Enforcing the substitution $\log x \to -x$ yields
$$\begin{align}
I(a)&=\int_{0}^{\infty} \frac{e^{-ax}-1}{x}\,e^{-x}\,dx\\\\
&=-\int_{0}^{\infty} \frac{e^{-(a+1)x}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/566475",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that $\sum_{n=0}^\infty (order\ {S_n})q^n=\prod_{m\ge 1}(1-q^m)^{-1}$ Let $T=\mathbb (C^*)^2$ acts on $\mathbb C[x,y]$ via $(t_1,t_2)(x,y)=(t_1x,t_2y)$, let $S_n$ be the set of ideals $I$ of $\mathbb C[x,y]$ such that $TI=I$ and $\mathbb C[x,y]/I$ is $n$-dimensional $\mathbb C$-vector space. If $order\ S_{0}=0$. s... | Since this looks like a homework problem let me just give an outline.
*
*Prove that an ideal is invariant under the torus action if and only if it is generated by monomials.
*Prove that monomial ideals $I \subset \mathbb C[x,y]$ can be identified with partitions. Hint: draw a square grid with squares $(i,j)_{i \geq... | {
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Finite extension of perfect field is perfect Let $E/F$ be a finite extension and $F$ be a perfect field.
Here, perfect field means $char(F)=0$ or $char(F)=p$ and $F^p=F$.
How to prove $E$ is also perfect field?
For $char(F)=0$ case, it's trivial, but for $char(F)=p$, no improvement at all...
Give me some hints
| Hint: Recall that a field is perfect if and only if every finite extension is separable. Now, if $L/E$ finite weren't separable, then clearly $L/F$ is finite and isn't separable.
| {
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Prime Number Theorem estimate Update
I have updated this question in light of the illuminating answers given below, which clearly point out my mistake. However, I still maintain that Legendre's original guess had some validity - it would seem, as the asymptotic starting point for the plot $\log x - \frac{x}{\pi(x)}$ as... | No, your value is a worse approximation. The true approximation/value of Legendre's constant is $1$.
By the prime number theorem, $$\pi(x)=\frac{x}{\log x}+\frac{x}{\log^{2}x}+O\left(\frac{x}{\log^{3}x}\right)=\frac{x}{\log x}\left(1+\frac{1}{\log x}+O\left(\frac{1}{\log^{2}x}\right)\right)$$ and so $$\frac{x}{\pi(x)}=... | {
"language": "en",
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Marking the point closest to each point We have $6000$ points in the plane. All distances between every pair of them are distinct.
For each point, we mark red the point nearest to it. What is the smallest number of points that can be marked red?
I divide the $6000$ points into $1000$ groups with $6$ points in each grou... | This is a known problem, and I use its standard dramatic interpretation to solve it. Consider $n$ marksmen standing in a field (so that all their pairwise distances are different). Each marksman simultaneously shoots and kills the closest marksman. What is the smallest number $k(n)$ of marksmen that can be killed?
Lem... | {
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The tangent to the curve $y=x^2+1$ at $(2,5)$ meets the normal to the curve at $(1, 2)$
Find the coordinates of the point where the tangent to the curve $y=x^2+1$ at the point $(2,5)$ meets the normal to the same curve at the point $(1,2).$
I tried to form 2 equations for each set of coordinates given, then solve the... | *
*First, you need to find $f'(x)= 2x$.
*Then to evaluate the slope of the tangent line at the point $(2, 5)$,
$m_1 = f'(2) = 4$. With slope $m_1=4$, and the point $(x_0, y_0) =
(2, 5)$, use the slope-point form of an equation to obtain the
equation of that tangent line.
$(y - y_0) = m_1(x - x_0)\tag{Point-Slop... | {
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A lot of terms to calculate lim I'm trying to prepare for exam and I came across a limit to calculate.
$$
\lim_{n->\infty} \frac{2^n + \left(1+\dfrac{1}{n^2}\right)^{n^3} + \dfrac {4^n}{n^4}}{\dfrac {4^n}{n^4} + n^3\cdot 3^n}
$$
When I'm trying to extract $4^n$ I end up with nothing.
And I managed to tell that $(1+\f... | $$\frac{2^n + e^n + \frac{4^n}{n^4}}{\frac{4^n}{n^4} + 3^nn^3} = \frac{2^nn^4 + e^nn^4 + 4^n}{4^n + n^73^n} = \frac{\frac{n^4}{2^n} + (\frac{e}{4})^n\cdot n^4 + 1}{1 + (\frac{3}{4})^nn^7} $$
Now, since $a^n$ is an infinite of magnitude greater than $n^b$ for all $b$, we conclude that all those fractions tends to zero (... | {
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Prove that there is an infinite number of rationals between any two reals I just stumbled upon this question: Infinite number of rationals between any two reals..
As I' not sure about my idea of a proof, I do not want to post this as an answer there, but rather formulate as a question.
My idea is as follows:
*
*$\ma... | Since $q_0$ has been found such that $a > q_0 > b$, you can use induction proof:
For all integer $n$, let $P(n)$ be :
there exist $q_0, \cdots, q_n \in \Bbb Q$ such that $a>q_0> \cdots >q_n >b$.
Then:
(i) $P(0)$ is true.
(2) Let us suppose $P(n)$ true for any $n\in \Bbb N$. Let $q_{n+1} \in \Bbb Q$ such that $ q_n >... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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DPLL Algorithm $ \rightarrow $ Resolution proof $ \rightarrow $ Craig Interpolation I really need help here for an exam that I got tomorrow ..
Let's say I got a bunch of constraints:
$
c1 = { \lnot a \lor \lnot b } \\
c2 = { a \lor c } \\
c3 = { b \lor \lnot c } \\
c4 = { \lnot b \lor d } \\
c5 = { \lnot c \lor \lnot d... | I'm not quite certain what your actual question is, but:
*
*The lower part of the page you scanned shows a resolution proof. I haven't checked every step, but at least the lower part looks correct.
*About choosing the variable sets in Craig interpolation: It is not possible to come up with these sets out of thin ai... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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probability and combinatorics mixed question A bus follows its route through nine stations, and contains six passengers. What is the probability that no two passengers will get off at the same station?
no detailed solution is required here but an idea of the general line of thought could be nice...
| This is an occupancy problem. You need to count the number of ways that 6 balls can get put into 9 sacks, such that each sack has at most 1 ball in it. Hint: since at most one person gets of at each bus stop, you are putting an order on the bus stops.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Expression for arbitrary powers of a particular $2\times2$ matrix Given$$\mathbf{M}=
\begin{pmatrix}
7 & 5 \\
-5 & 7 \\
\end{pmatrix}
$$, what's the formula matrix for $\mathbf{M}^n$? The eigenvalues and eigenvectors are complex and need to generate a real number formulas for each com... | $\textbf M=\begin{bmatrix}7&5\\-5&7\end{bmatrix}$
The eigenvalues will be the roots of the characteristic polynomial, $\lambda^2-(\mathrm{tr}\ \textbf M)\lambda+\det\textbf M=0$. $\lambda^2-14\lambda+74=0$, so $\lambda=7\pm5i$.
Thus, the diagonalization of $\textbf M=\textbf A^{-1}\cdot\begin{bmatrix}7+5i&0\\0&7-5i\en... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove the estimate $|u(x,t)|\le Ce^{-\gamma t}$ Assume that $\Omega \subset \Bbb R^n$ is an open bounded set with smooth boundary, and $u$ is a smooth solution of
\begin{cases}
u_t - \Delta u +cu = 0 & \text{in } \Omega \times (0, \infty), \\
u|_{\partial \Omega} = 0, \\
u|_{t=0} = g
\end{cases}
and the function $C$ s... | Let me elaborate on my comment that it is indeed possible to do this via energy methods. Let $p \geq 2$ be an even integer and differentiate under the integral and apply the chain rule to get
$$\partial_t \|u(t)\|_p^p = \int_\Omega \! p|u|^{p-1}u_t \, dx = p\int_\Omega \!|u|^{p-1}(\Delta u - cu) \, dx$$
Now integrate b... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Symbol for "is closest to"? I am writing a paper on probabilities and we have to find a $k$ such that $P_n(k)$ is "closest to" $P_0$.
$P_0$ is getting 4-of-a-kind in a five card hand in a standard 52 card deck.
$P_n(k)$ is probability of getting $k$-of-a-kind in an $n$ card hand in some modified 88 card deck.
I want... | You could either say exactly what you said above, or, more formally: $|P_i-P_n(k)|$ is minimized for $i=0$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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$1 + \frac{1}{1+2}+ \frac{1}{1+2+3}+ ... + \frac{1}{1+2+3+...+n} = ?$ How do I simplify the following series
$$1 + \frac{1}{1+2}+ \frac{1}{1+2+3}+ \frac{1}{1+2+3+4} + \frac{1}{1+2+3+4+5} + \cdot\cdot\cdot + \frac{1}{1+2+3+\cdot\cdot\cdot+n}$$
From what I see, each term is the inverse of the sum of $n$ natural numbers.
... | HINT: $$\frac {2} {n(n+1)} = \frac 2 n - \frac 2 {n+1}$$
| {
"language": "en",
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Solving for the zero of a multivariate
How does one go about solving the roots for the following equation
$$x+y+z=xyz$$
There simply to many variables. Anyone have an idea ?
| If we fix one of the variables, we get a hyperbola in that plane. So, for example, fixing any $z = z_0,$ this is your relationship:
$$ \left(x - \frac{1}{z_0} \right) \left(y - \frac{1}{z_0} \right) = \; 1 + \frac{1}{z_0^2} $$
Makes me think the surface could be connected.
Indeed, as $|z| \rightarrow \infty,$ the cu... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving that if these quadratics are equal for some $\alpha$, then their coefficients are equal Let
$$P_1(x) = ax^2 -bx - c \tag{1}$$
$$P_2(x) = bx^2 -cx -a \tag{2}$$
$$P_3(x) = cx^2 -ax - b \tag{3}$$
Suppose there exists a real $\alpha$ such that
$$P_1(\alpha) = P_2(\alpha) = P_3(\alpha)$$
Prove
$$a=b=c$$
Equating ... | Denote
$$Q_1(x)=P_1(x)-P_2(x)=(a-b)x^2-(b-c)x-(c-a);$$
$$Q_2(x)=P_2(x)-P_3(x)=(b-c)x^2-(c-a)x-(a-b);$$
$$Q_3(x)=P_3(x)-P_1(x)=(c-a)x^2-(a-b)x-(b-c).$$
Then $\alpha$ is a real root of the equations $Q_i(x);$ so that $\Delta_{Q_i(x)}\geq 0 \ \ \forall i=1,2,3;$ where $\Delta_{f(x)}$ denoted the discriminant of a quadrati... | {
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Rudin Example 3.35B Why the $n$th root of $a_n$ is $1/2$? For Baby Rudin Example 3.35(b), I understand how the $\liminf$ and $\limsup$ of the ratio test were found, but I am not clear why $\ \lim \sqrt[n]{a_n } = \frac{1}{2} $.
Please help.
| The sequence in question is
$$\frac{1}{2} + 1 + \frac{1}{8} + \frac{1}{4}+ \frac{1}{32}+ \frac{1}{16}+\frac{1}{128}+\frac{1}{64}+\cdots$$
In case the pattern is not clear, we double the first term, the divide the next by $8$, the double, then divide by $8$, and so on.
The general formula for an odd term is $a_{2k-1}=\... | {
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Prove that any line passing through the intersection of two bisectors is also a bisector. Given an arbitrary closed shape $F$, a line, $H$, that bisects $F$ horizontally, and a line, $V$, that bisects $F$ vertically, is it true that any line that passes through the intersection of $H$ and $V$ also bisects $F$?
I know t... | The lines which bisect the area of a triangle form an envelope as shown in this picture
The blue medians intersect in the centroid of the triangle, but no other lines through the centroid bisect the area of the triangle; none of the green bisectors of the area of the triangle pass through the centroid.
It is easiest t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Countability (show set is countable) Show that the set $\mathbb{Z}_+\times\mathbb{Z}_+$ is countable.?
To solve this you have to show a one to one correspondence.
$\mathbb{Z}_+\times\mathbb{Z}_+\to\mathbb{Z}_+$ Then my book recommends using $f(m,n) = 2^m\times3^n$ (or any other primes) to show it is one to one.
Where ... | It isn't important that you use the primes 2 or 3. The fact that there is a one to one correspondence between $\mathbb{Z}_+ \times \mathbb{Z}_+$ and $f(\mathbb{Z}_+,\mathbb{Z}_+)$ is a consequence of the unique factorization in $\mathbb{Z}$. The map
$$f: \mathbb{Z}_+ \times \mathbb{Z}_+ \to f(\mathbb{Z}_+,\mathbb{Z}_+)... | {
"language": "en",
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Value of sum of telescoping series $$\sum_{n\geqslant1}\frac{1}{\sqrt{n}} -\frac{ 1}{\sqrt {n+2}}$$
In looking at the first five partial sums, I am not convinced the series is telescopic (the middle terms don't cancel out).
Thanks in advance!
| $\sum_{k=1} ^n \frac{1}{\sqrt{k}}- \frac{1}{\sqrt{k+2}}$
$= \frac{1}{\sqrt{1}}- \frac{1}{\sqrt{3}}+ \frac{1}{\sqrt{2}}- \frac{1}{\sqrt{4}} +\frac{1}{\sqrt{3}}- \frac{1}{\sqrt{5}}+ \frac{1}{\sqrt{4}}- \frac{1}{\sqrt{6}}+ \frac{1}{\sqrt{5}}- \frac{1}{\sqrt{7}}+...+ \frac{1}{\sqrt{n-2}}- \frac{1}{\sqrt{n}}+ \frac{1}{\sqrt... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Can you factor out vectors? My prof introduced eigenvalues to us today:
Let $A$ be an $n \times n$ matrix. If there a scalar $\lambda$ and an $n-1$ non-zero column vector $u$, then
$$Au = \lambda u$$
then $\lambda$ is called an eigenvalue and $u$ is called an eigenvector.
$$Au - \lambda u = 0$$
$$\implies (A - \lambda... | The distributive laws apply to matrix (or matrix-vector) multiplication.
$$\eqalign{(A+B) C &= AC + BC\cr
A(C+D) &= AC + AD\cr}$$
whenever $A$,$B$,$C$,$D$ have the right dimensions for these to make sense.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How to find all vectors so that a vector equation can be solved? Unfortunately, my text book doesn't clarify this process at all. It's asking to find all all vectors [a b] so that the vector equation can be solved. The vector equation is:
$c1$ $[3,1]$ + $c2 [6,2]$=$[a, b]$
The linear system would look like:
$3c1+6c2=... | Hint:
You have two equations, and two variables (we treat a, b as constants). Set up the associated augmented coefficient matrix, row reduce, and solve for $c_1, c_2$, which can each be expressed as functions of $a, b$. From that, you should also be able to express $a, b$ as functions of the constants $c_1, c_2$.
Asso... | {
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Recurrence sequence limit I would like to find the limit of
$$
\begin{cases}
a_1=\dfrac3{4} ,\, & \\
a_{n+1}=a_{n}\dfrac{n^2+2n}{n^2+2n+1}, & n \ge 1
\end{cases}
$$
I tried to use this - $\lim \limits_{n\to\infty}a_n=\lim \limits_{n\to\infty}a_{n+1}=L$, so $L=\lim \limits_{n\to\infty}a_n\dfrac{n^2+2n}{n^2+2n+1}=L\cdot... | Put $a_n=\frac{n+2}{n+1}b_n$. Then $b_n=b_1=\frac{1}{2}$, so $a_n=\frac{n+2}{2n+2}$ and $\lim \limits_{n\to\infty}a_n=\frac{1}{2}$.
| {
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"question_score": "3",
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"answer_id": 3
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Let $f:\mathbb{R}^2\to\mathbb{R}$ be a $C^1$ function. Prove that the restriction is not injective. Let $f:\mathbb{R}^2\to\mathbb{R}$ be a $C^1$ function. And let $D$ an open subset of $\mathbb{R}^2$. Prove that the restriction of $f$ to $D$ is not injective.
Im trying to solve this but i dont know how...
The problem h... | Assume by contradiction that $f_D$ is injective.
Let $U \subset D$ be an open connected subset of $D$. Then $f|_U$ is also injective. As $U$ is connected then $f(U)$ is connected, hence an interval. Let $d \in f(U)$ be an interior point.
As $f$ is injective, there exists an unique $e \in U$ so that $f(e)=d$.
Then $f$ ... | {
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Evaluating $\int \frac{\sin^2(x)}{\sqrt{\cos(x)}} \mathrm dx$ I would like to get some advice how to evaluate the integral,
$$\int \frac{\sin^2{x}}{\sqrt{\cos{x}}} \mathrm dx$$
| Integration by parts, with $g(x)=\sin x$, and $$f'(x)=\frac{\sin x}{\sqrt{\cos x}}=-2\cdot\frac{\cos'x}{2\cdot\sqrt{\cos x}}=-2\cdot(\sqrt{\cos x})'\iff f(x)=-2\cdot\sqrt{\cos x}$$ then recognizing the expression of the incomplete elliptic integral of the first kind in $\int f(x)g'(x)dx$
| {
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Take Laplace Transform of the integral J_0 I was just wondering how to use tables from Spiegal to solve $\int_0^\infty J_0(2\sqrt{ut}) J_0(u) du$ At the moment, I see similar transforms on page 244, but I don't actually know how to combine the laplace transforms of the first $J_0$ and the second $J_0$
Any help is appre... | Well, I have a way to do it without referring to the tables, but instead using two well-known representations of a Bessel function. First write
$$J_0(u) = \frac{1}{\pi} \int_0^{\pi} d\theta \, e^{i u \cos{\theta}}$$
Then assuming we may interchange the order of integration, the integral is equal to
$$\begin{align}\int... | {
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Is there a continuous function $f:S^1 \to \mathbb R$ which is one-one? Is there a continuous function $f:S^1 \to \mathbb R$ which is one-one?
| Suppose such a function exists. Let $u : [0,1] \to S^1$ be a suitable path that traces around the circle, and consider $g = fu$. This $g : [0,1] \to \mathbb R$ is one-to-one except that $g(0) = g(1)$.
Consider $y = g(1/2)$. It must be either greater than or less than $g(0)$. Pick some value $z$ between $g(0)$ and $y$. ... | {
"language": "en",
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Finding $\operatorname{Ext}^{1}(\Bbb Q,\Bbb Z)$
I am trying to compute $\operatorname{Ext}^{1}(\Bbb Q,\Bbb Z)$ explicitely.
Using $\Bbb Q/\Bbb Z$ I constructed a natural injective resolution of $\Bbb Z$, and I know that $\Bbb Q/\Bbb Z$ is injective. Please help after that.
| We have the following exact sequence: $$\operatorname{Hom}_{\Bbb Z}(\Bbb Q,\Bbb Q)\to \operatorname{Hom}_{\Bbb Z}(\Bbb Q,\Bbb Q/\Bbb Z)\to\operatorname{Ext}^1_{\Bbb Z}(\Bbb Q,\Bbb Z)\to\operatorname{Ext}^1_{\Bbb Z}(\Bbb Q,\Bbb Q).$$ Since $\operatorname{Hom}_{\Bbb Z}(\Bbb Q,\Bbb Q)\cong\Bbb Q$ and $\operatorname{Ext}^1... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Orthogonal Projection of a matrix Let $V$ be the real vector space of $3 \times 3$ matrices with the bilinear form $\langle A,B \rangle=$ trace $A^tB$, and let $W$ be the subspace of skew-symmetric matrices. Compute the orthogonal projection to $W$ with respect to this form, of the matrix $$\begin{pmatrix} 1& 2 & 0\\ 0... | Find the orthogonal complement to $W$, i.e.
$$W^{\perp} := \{ X \in V : \langle X, W \rangle = 0\}$$
Assuming that $\dim W + \dim W^{\top} = \dim V$, we can write any $X \in V$ as a linear combination:
$$X = \alpha A + \beta B$$
where $\alpha,\beta \in \mathbb{R}$, $A \in W$ and $B \in W^{\top}$. The orthogonal projec... | {
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Finding the interval convergence power series? How would I find the interval of convergence of this power series?
$\sum\frac{1x^k}{k^22^k}$
I performed the ratio test and did.
$\frac{x^{k+1}}{(k+1)^2(2)^{k+1}}$*$\frac{k^2(2^k)}{x^k}$
Then I got
$k\rightarrow\infty$
$x\frac{k^2}{2(k+1)^2}$
$-1<\frac{1}{2}x<1$
$x=2$
$x=-... | The series converges for $x=2$. You may ask why...
Why not use the integral test, i.e.
$\lim_{n\rightarrow \infty} \int_{1}^{n} x^{-2}\; dx = \lim_{n \rightarrow \infty} 1 - \frac{1}{n} = 1$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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No idea how to prove this property about symmetric matrices This is from homework, so please hints only.
Suppose $A$ is symmetric such that all of its eigenvalues are 1 or -1. Prove that $A$ is orthogonal.
The converse is really easy, but I really have no idea how to do this. Any hints?
| Edited Saturday 16 November 2013 10:03 PM PST
Well, it seems the "hints" have had their desired effect, so I'm editing this post to be an answer, pure and simple.
That being said, try this:
since $A$ is symmetric, there exists orthogonal $O$ such that $O^TAO = \Lambda$, with $\Lambda$ diagonal and $\Lambda_{ii} = \pm 1... | {
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How to separate a partial differential equation where R is a function of three variables? Using the method of separation of variables, how can I separate each X,Y,Z if the differential equation has a function of R(x,y,z)?
Example:
$ R_{xx} + R_{yy} + R_{zz} = 0 $
I understand how to apply the method if R is only a fun... | The argument parallels the two variable case. Setting
$R(x, y, z) = X(x)Y(y)Z(z), \tag{1}$
we have
$X_{xx}(x)Y(y)Z(z) + X(x)Y_{yy}(y)Z(z) + X(x)Y(y)Z_{zz}(z) = 0, \tag{2}$
and dividing through by $X(x)Y(y)Z(z)$ we obtain
$X_{xx} / X + Y_{yy} / Y + Z_{zz} / Z = 0, \tag{3}$
which we write as
$X_{xx} / X = -Y_{yy} / Y - ... | {
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Implicit differentiation I want to differentiate
$x^2 + y^2=1$ with respect to $x$.
The answer is $2x +2yy' = 0$.
Can some explain what is implicit differentiation and from where did $y'$ appear ? I can understand that $2x +2yy' = 0$ is a partial derivative but then it becomes multi calc not single.
This is in a chapt... | The function $x^2 + y^2 = 1$ defines $y$ implicitly as a function of $x$. In this case, we have $y^2 = 1 - x^2$. Thus, instead or writing $y$ in the equation we can write $f(x)$ where $f(x)^2 = 1 - x^2$. This leaves the problem of differentiating $x^2 + f(x)^2 = 1$. In this form, we can see how to apply the chain rule
... | {
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"timestamp": "2023-03-29T00:00:00",
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How to prove that a group of order $72=2^3\cdot 3^2$ is solvable? Let $G$ be a group of order
$$72=2^3\cdot 3^2$$
Without using Burnside's Theorem, how to show that $G$ is solvable?
Atempt:
If we can show that $G$ has at least one non-trivial normal subgroup $N$, then it would be easy to show it is solvable. Indeed,
$$... | If $G$ has 4 Sylow-3 subgroups, $G$ acts on those subgroups via conjugation, inducing a homomorphism $G\to S_4$. Since $|S_4|=24<72=|G|$, this map must have a non-trivial kernel. If the morphism is not the trivial map, you are done. What can you say if the kernel is all of $G$?
| {
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"timestamp": "2023-03-29T00:00:00",
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$\sqrt x$ is uniformly continuous Prove that the function $\sqrt x$ is uniformly continuous on $\{x\in \mathbb{R} | x \ge 0\}$.
To show uniformly continuity I must show for a given $\epsilon > 0$ there exists a $\delta>0$ such that for all $x_1, x_2 \in \mathbb{R}$ we have $|x_1 - x_2| < \delta$ implies that $|f(x_1) -... | In $[0,1]$ we want
$$
\begin{split}
|f(x)-f(y)|=&\frac {|x-y|}{\sqrt{x}+\sqrt{y}} <\varepsilon\\
\Updownarrow\\
|x-y|<&\varepsilon(\sqrt{x}+\sqrt{y})<2\varepsilon\:\:\text{ so }\:\:\delta=2\varepsilon
\end{split}
$$
In $[1, \infty]$,
$$
\frac {|x-y|}{\sqrt{x}+\sqrt{y}} < |x-y|\:\:\text{ so }\:\:\delta=2\varepsilo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/569928",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "46",
"answer_count": 6,
"answer_id": 4
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Elementary properties of integral binary quadratic forms Let $f = ax^2 + bxy + cy^2$ be a binary quadratic form over $\mathbb{Z}$.
$D = b^2 - 4ac$ is called the discriminant of $f$.
We say $f$ is positive definite if $a \gt 0$ and $D \lt 0$(cf. this question).
We say $f$ is primitive if gcd$(a, b, c) = 1$.
Let $\sigma ... | 1) A binary quadratic form $f(x, y)=ax^2+bxy+cy^2$ can be written
$$f(x, y)=\begin{pmatrix} x & y \end{pmatrix} \begin{pmatrix} a & \frac{b}{2} \\ \frac{b}{2} & c \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix}$$
so $f(x, y)$ corresponds to the $2 \times 2$ matrix $M$ in the center, with discriminant $D=-4\det M$.
N... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How prove this $a_{n}>1$ let $0<t<1$, and $a_{1}=1+t$, and such
$$a_{n}=t+\dfrac{1}{a_{n-1}}$$
show that $a_{n}>1$
My try: since
$$a_{1}=1+t>1$$
$$a_{2}=t+\dfrac{1}{a_{1}}=t+1+\dfrac{1}{1+t}-1>2\sqrt{(t+1)\cdot\dfrac{1}{1+t}}-1=2-1=1$$
$$a_{3}=t+\dfrac{1}{a_{2}}=t+\dfrac{1}{t+\dfrac{1}{t+1}}=t+\dfrac{t+1}{t^2+t+1}=1+\... | Let $\displaystyle \mu = \frac{t + \sqrt{t^2+4}}{2}$, we have
$$\mu > 1\quad\text{ and }\quad\mu(t - \mu) = \left(\frac{t + \sqrt{t^2+4}}{2}\right)\left(\frac{t - \sqrt{t^2+4}}{2}\right) = -1$$
From this, we get
$$a_{n+1} - \mu = t - \mu + \frac{1}{a_n} = \frac{1}{a_n} - \frac{1}{\mu} = \frac{\mu - \alpha_n}{\mu a_n}$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/570116",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
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If $G$ is a group and $N$ is a nontrivial normal subgroup, can $G/N \cong G$? I know $G/N$ is isomorphic to a proper subgroup of $G$ in this case, so the gut instinct I had was 'no'. But there are examples of groups that are isomorphic to proper subgroups, such as the integers being isomorphic to the even integers, so ... | Yes. Let $G$ be the additive group of the complex numbers, and let $N$ be the subgroup consisting of the real numbers.
Edit in response to comment by @GA316:
$(\mathbb C,+)/\mathbb R$ is clearly isomorphic to $(\mathbb R, +)$, and it is well known (but this requires the Axiom of Choice) that $(\mathbb C,+)\cong(\mathbb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/570177",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 6,
"answer_id": 0
} |
How many rooted plane trees tn are there with n internal nodes? How many rooted plane trees tn are there with n internal nodes? Plane means that left and
right are distinguishable (i.e. mirror images are distinguishable), and rooted simply means
that the tree starts with a single root. For the sake of understanding, th... | In graph theory and more specifically in rooted plane trees there is a fundamental sentence: The number of rooted plane trees with n nodes equals to n-th Catalan number, that is |Tn| = Cn.
I hope to have helped you.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/570288",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finding the limit $ \lim_{x\to 0}\frac{(1-3x)^\frac{1}{3} -(1-2x)^\frac{1}{2}}{1-\cos(\pi x)}$ I cannot find this limit:
$$
\lim_{x\to 0}\frac{(1-3x)^\frac{1}{3} -(1-2x)^\frac{1}{2}}{1-\cos(\pi x)}.
$$
Please, help me.
Upd: I need to solve it without L'Hôpital's Rule and Taylor expansion.
| Write it as $\left(\frac{\cos\pi x-\cos0}{x-0}\right)^{-1}\times\frac{\left(1-3x\right)^{\frac{1}{3}}-\left(1-2x\right)^{\frac{1}{2}}}{x-0}$.
Then $\lim_{x\rightarrow0}\frac{\cos\pi x-\cos0}{x-0}$ can be recognized
as $f'\left(0\right)$ for $f\left(x\right)=\cos\pi x$
and $\lim_{x\rightarrow0}\frac{\left(1-3x\right)^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/570406",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Taylor/Maclaurin Series Show that if x is small compared with unity, then $$f(x)=\frac{(1-x)^\frac{-2}{3}+(1-4x)^\frac{-1}{3}}{(1-3x)^\frac{-1}{3}+(1-4x)^\frac{-1}{4}}=1-\frac{7x^2}{36}$$
In my first attempt I expanded all four brackets up to second order of x, but this didn't lead me to something that could be express... | The development of the last fraction (ratio of the two quadratic polynomials) is
1 - 7 x^2 / 36 + 7 x^3 / 36 + 35 x^4 / 144.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/570497",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Evaluate the integral $\int_{\gamma}\frac{z^2+2z}{z^2+4}dz$ Evaluate $$\int_{\gamma}\frac{z^2+2z}{z^2+4}dz$$ where the contour $\gamma$ is
1.) the circle of radius $2$ centered at $2i$, traversed once anti-clockwise.
2.) the unit circle centered at the origin, traversed once anti-clickwise.
So here we would have to use... | $(1)$ Apply the residue theorem. $\int_{\gamma}\frac{z^2+2z}{(z-2i)(z+2i)}dz=2i\pi(\sum res_{z=z_k})$. Define $z_0:=z+2i.$ Thus, $res_{z=z_0}f(z)=\frac{z_k^2+2z_k}{2z_k}$, for $k=0$.
$(2)$ Notice that none of your singluar points are in your contour $\Rightarrow $$\int_{|z|=1}\frac{z^2+2z}{(z-2i)(z+2i)}dz=0$, by Cauchy... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/570572",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Irreducible polynomial $f$ as quotient - effect on $\mathbb{Z}_5[x]$ I want to get a better understanding of quotient rings so I have two questions.
Let $f(x) = x^2 + 2$
Let $R = \mathbb{Z}_{5}/(f(x))$
Now as $f$ is irreducible in $\mathbb{Z}_{5}$ we have that $R$ is a field with elements being all polynomials in $\mat... | In $R$: $x^2 + 2 = 0$, so $(2x)x = 2 x^2 = -4 = 1$, i.e., $x^{-1} = 2x$.
For the second question, $R$ does not have the same elements as $S$; $R$ and $S$ only happen to have the same number of elements and they can be represented by the same elements of ${\mathbb Z}_5[x]$, but that's it. Now because $x^2 + 1$ is reduci... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/570643",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
$\gcd$ of polynomials over a field I have the polynomials $f,g\neq 0 $ over a field $F$. We know also that $\gcd(f,g)=1$ and
$$
\det \begin{pmatrix}
a & b \\
c & d \\
\end{pmatrix}\neq 0.
$$
I need to prove that $\gcd(af+bg,cf+dg) = 1 $ for every $a,b,c,d \in F$.
I really do not know ho... | Just work through the equations of gcd (bearing in mind that you're working in $f(x)$, hence constants do not matter):
$ \gcd(af+bg, cf+dg) = \gcd(adf+bdg, cbf + bdg) = \gcd( (ad-bc)f, cbf + bdg) = \gcd( f, cbf+bdg) = \gcd(f, bdg) = \gcd(f, g) = 1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/570750",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
About sum of three squares I am trying to find those $k$ for which the expression $1+(10k+4)^2 +(10m+8)^2$ is never a square number for any $m$.
Thank you!
| You are trying to solve $J + (10m+8)^2 = n^2$ or show that no solution exists, where $J=1+(10k+4)^2$. For any $J$, you can solve $J = n^2 - p^2$ by writing it as $J = (n-p)(n+p)$ and then finding all factorizations of $J$ into two factors of equal parity (both even or both odd). This gives you all possible choices for ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/570845",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Velocity of a Particle Consider a particle moving in a straight line from the fixed point $(0,2)$ to another $(\pi,0)$. The only force acting on the particle is gravity.
How would we parametrically define the motion of the particle with time?
From kinematics, I found that
$\hspace{150pt} y(t)=2-\dfrac{gt^2}{2}$
The s... | $\newcommand{\+}{^{\dagger}}%
\newcommand{\angles}[1]{\left\langle #1 \right\rangle}%
\newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}%
\newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}%
\newcommand{\dd}{{\rm d}}%
\newcommand{\isdiv}{\,\left.\right\vert\,}%
\newcommand{\ds}[1]{\displaystyle{#1}}%
\new... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/570912",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Is Hoeffding's bound tight in any way? The inequality:
$$\Pr(\overline X - \mathrm{E}[\overline X] \geq t) \leq \exp \left( - \frac{2n^2t^2}{\sum_{i=1}^n (b_i - a_i)^2} \right) $$
Is this bound (or any other form of hoeffding) tight in any sense? e.g. does there exist a distribution for which the bound is no more than ... | Because it fully answers the question, I will quote almost verbatim Theorem 7.3.1
from Matoušek, Jiří, and Jan Vondrák. "The probabilistic method." Lecture Notes, Department of Applied Mathematics, Charles University, Prague (2001). downloadable as of today at http://www.cs.cmu.edu/~15850/handouts/matousek-vondrak-prob... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/571032",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 2,
"answer_id": 0
} |
How to solve equation $\tau(123)(45)(67)\tau^{-1}=(765)(43)(21)$ and alike? Let $\sigma\in S_7$ be $(123)(45)(67)$. And Find $\tau\in S_7$ such that $\tau\sigma\tau^{-1}=(765)(43)(21)$. I understand that in symmetric group conjugate elements have the same cycle structure. Hence, $\tau$ should share the structure of $\s... | There are more solutions than just $\tau=(17)(26)(35)$. Your task was only to find one such $\tau$, so that's fine, but it is not too hard to find them all. And in the process, see that the cycle structure of $\tau$ is unrelated to the cycle structure of your permutations.
The cycles in your given permutations are all ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/571076",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Showing that a sequence of random variables with $\mathcal{L}(X_{n})$ (I.e., law) uniform on $[-n,n]$ does not converge at all Let $(X_{n})_{n\geq 1}$ be a sequence of real valued random variables with $\mathcal{L}(X_{n})$ (that is, law or distribution) uniform on $[-n,n]$. In what sense(s) do $X_{n}$ converge to a ran... | Convergence in distribution means
$$\lim_{n\rightarrow \infty}F_{X_n}(x) = F_X(x)$$
where the RHS is a distribution function, and the equality to hold for every $x$ for which $F_X(x)$ is continuous.
Our distribution function is
$$F_{X_n}(x)= \begin{cases}0&\text{if $x<-n$}\\\frac{x+n}{2n}&\text{if $x\in[-n,n]$} \\1&... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/571180",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
$\mathbb{E}[e^{Xt}] = \mathbb{E}[\mathbb{E}[e^{Xt}\mid Y]] = \mathbb{E}[M_{X\mid Y}(t)]$? $$\mathbb{E}[e^{Xt}] = \mathbb{E}[\mathbb{E}[e^{Xt}\mid Y]] = \mathbb{E}[M_{X\mid Y}(t)]$$
How do I get the above statement? I don't understand how in the 1st step $e^{Xt}=\mathbb{E}[x^{Xt}\mid Y]$ then in the 2nd $\mathbb{E}[e^{X... | The first step is what is called the law of Iterated Expecations. Simply put, if $X,Y$ are random variables then, $$\mathbb{E}_X[X] = \mathbb{E}_Y[\mathbb{E}_{X \mid Y}[X \mid Y]]$$ and by definition, $\mathbb{E}_{X \mid Y}[X \mid Y]$ is a function of $Y$ and again by definition $\mathbb{E}_{X \mid Y}[e^{xt}\mid Y]$ is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/571264",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
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