Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
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Discrete Math/Calculus (Given summation formula, find value of x) Textbook Question:
Given $\sum\limits_{i=0}^\mathbb{50}{50\choose i}8^i $ = $x^{100}$ . Find the values of x ?
Textbook Solution:
$\sum\limits_{i=0}^\mathbb{50}{50\choose i}8^i $ = $(1+8)^{50} = 9^{50} = [(+-3)^2)]^{50}$
$\therefore$ x = +-3
Personal Lo... | The book is using Binomial Theorem:
$$\sum_{i=0}^{n}{n \choose i}a^ib^{n-i}=(a+b)^{n}$$
Use $a=8$, $b=1$ and $n=50$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2153435",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to show that if $9n^2=a^2+b^2$, $a$ and $b$ are multiples of $3$ To be honest, I don't know where to start with this problem:
Let $n\in \mathbb{N}$. Prove that if $9n^2$ is the sum of two perfect squares $(a^2,b^2)$, then $a$ and $b$ are multiples of $3$.
| If one of them is not, say wlog $ a $, then $ a^2 + b^2 \equiv 0 \pmod{9} $, and $ (b/a)^2 \equiv -1 \pmod{9} $ so that $ b/a $ has order $ 4 $ in the group $ (\mathbf Z/9 \mathbf Z)^{\times} $ of order $ 6 $, contradicting Lagrange's theorem.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2153527",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Why can't we define "the" derivative when mapping from $\mathbb{R}^2 \to \mathbb{R}$? When we map from $\mathbb{R} \to \mathbb{R}$, the derivative is given by $\lim_{x \to a} \dfrac {f(x) - f(a)}{x-a}.$ When we map from $\mathbb{C} \to \mathbb{C}$ (basically $\mathbb{R}^2 \to \mathbb{R}^2$) we also have a definition f... | One way to look at is that, as you say, there's no way to "divide" vectors in $\mathbf{R}^2$ that really makes sense.
But you can think of it this way. When you differentiate a function $y = f(x)$ at a point $x_0$, what you're doing is saying that that function can be approximated by a linear function for values of $x$... | {
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"url": "https://math.stackexchange.com/questions/2153633",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
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Union of Intersections I am trying to prove that $S\cup(S\cap T)=S$ and the dual statement $S\cap(S\cup T)=S$ for a class, and have gotten stuck with my proof.
$$
S\cup(S\cap T)
$$
$$
=\{x|x\in S\lor x\in (S\cap T)\}
$$
$$
=\{x|x\in S\lor (x\in S\land ]x\in T)\}
$$
$$
=\{x|(x\in S\lor x\in S)\land (x\in S\lor x\in T)\... | We have
$$S\cup(S\cap T)$$
Let $R = S\cap T$
We know $R\subseteq S$ since the intersection of some set $S$ with any other set is a subset of $S$.
We also know the union of $S$ and any subset of $S$--including $R$--is $S$, i.e. $R\cup S=S$.
Therefore, $S\cup(S\cap T)=S$
| {
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"timestamp": "2023-03-29T00:00:00",
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How to convert (B ∨ ¬A) → (C ∨ ¬D) to conjunctive normal form? I have tried solving it. But, I am stuck:
(B ∨ ¬A) → (C ∨ ¬D) // removing implication
¬(B ∨ ¬A) V (C ∨ ¬D) // using De Morgan's Laws
(¬B ∧ A) V (C ∨ ¬D) // This is not CNF. What to do next?
| $(B \lor \lnot A) → (C \lor \lnot D)\label{1}\tag{1}$
$(\lnot B \land A) \lor C \lor \lnot D\label{2}\tag{2}$
The best way to answer these types of questions is by Karnaugh Map. If you are not familiar with them there is a page here that explains them quite well
https://www.facstaff.bucknell.edu/mastascu/eLessonsHTM... | {
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"url": "https://math.stackexchange.com/questions/2153865",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Proving solutions of Linear Variational Problem Let $$a(v,w) = \int_0^1 a_o(x)v(x)w(x)dx + \lambda \int_0^1 v(y)w(x)dxdy, \; \; \forall v,w \in L^2(0,1)$$ be a bilinear, continuous, elliptic functional with $a_0(x) \in C^0[0,1], \; \;a_0(x) > 0$ on $[0,1]$ and $\lambda \in \mathbb{R}.$ Suppose that $f \in L^2(0,1)$ and... | i) integrate $u(x) + \lambda \int_{0}^{1} u(y) dy = f(x)$ from $0$ to $1$ to get
\begin{equation}
\int_{0}^{1} u(x) dx + \lambda \int_{0}^{1} \int_{0}^{1} u(y) dy dx = \int_{0}^{1} f(x) dx
\end{equation}
now giving $\int_{0}^{1} u(y) dy$ a name, say, $\int_{0}^{1} u(y) dy = q$, we have
\begin{equation}
q + \lambda \int... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2153959",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Partial derivatives turning into regular derivative If $z=z(x,y)$, and $x=x(t),y=y(t)$, write down the expression for $\frac{dz}{dx}$.
I am confused by the difference between partial and regular derivatives when they are used together.
| Observe that
$$dz=\frac{\partial z}{\partial x}dx+\frac{\partial z}{\partial y}dy$$
$$\implies \frac{dz}{dx}=\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}\cdot \frac{dy}{dx}=\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}\cdot \frac{dy}{dt}\cdot \frac{dt}{dx}$$
$$\implies \frac{dz}{dx}=\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2154104",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solve the equation : $\tan \theta + \tan 2\theta + \tan 3\theta = \tan \theta \tan 2\theta \tan 3\theta $ I've been having some trouble solving this equation. (The solution in my book is given as $ \frac {n \pi}{3}, n \in Z $)
Here is what I've done
$$\frac {\sin \theta}{\cos \theta} + \frac {\sin 2\theta} {\cos 2\thet... | We have
$$\tan (A+B+C) = \frac{\tan A + \tan B + \tan C - \tan A \tan B \tan C}{1-(\tan A \tan B + \tan B \tan C + \tan C \tan A)}$$
and the given condition implies that $\tan (6\theta) = 0$. Thus $\theta = \frac{n\pi}{6}$, $n \in \mathbb{Z}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2154221",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 3
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How to take the integral? $\int \frac{x^2-3x+2}{x^2+2x+1}dx$ $$\int \frac{x^2-3x+2}{x^2+2x+1}dx$$
So after all I had
$$ \frac{-5x+1}{(x+1)^2} = \frac{A}{(x+1)} + \frac{B}{(x+1)^2}$$
and of course $$ \int xdx $$
but it is easy to solve, I do not know how to act with devided things, probably solve the system, or is there... | \begin{align*}
\frac{x^2 - 3x + 2}{x^2 + 2x + 1} &= 1 + \frac{1-5x}{(x+1)^2} \\
&= 1 + \frac{A}{x+1} + \frac{B}{(x+1)^2}
\end{align*}
Then
\begin{align*}
\frac{1-5x}{(x+1)^2} &= \frac{A}{x+1} + \frac{B}{(x+1)^2} \\
&= \frac{A(x+1)+B}{(x+1)^2} \\
1-5x &= Ax+(A+B)
\end{align*}
So $A = -5$ and $B=6$.
Therefore
\begin{ali... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Factorise an ideal into a product of prime ideals So for the ideal $I = (20+\sqrt{-5})_{R}$ when $K=\mathbb{Q}(\sqrt{-5})$, how do I factorise this into a product of prime ideals. Do you start by taking the norm of $I$ and decomposing it into a product of primes? So
$$N(I) = 405 = 3^{4} \times 5.$$
If so, where do I go... | Recall that the primes $ 3, 5 $ split as $ 3R = \mathfrak p \mathfrak p' $ and $ 5R = (\sqrt{-5})^2 $ in $ R = \mathcal O_K = \mathbf Z[\sqrt{-5}] $. The norm suggests that the ideal $ I = (20 + \sqrt{-5}) $ factors as $ \mathfrak p^i \mathfrak p'^j (\sqrt{-5}) $, where $ i + j = 4 $. It follows upon division that
$$ \... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Difference between two products Let $q$ be a square free natural number. Can the difference
$$
\prod_{p \mid q} (p^2+1) - \prod_{p \mid q} (p-1)^2
$$
be estimated in terms of $q$? What would be the correct order of the difference in terms of $q$? Is the difference $\asymp q^2$? Can something be said about
$$
\prod_{p \... | Such difference equals
$$ q^2\left[\prod_{p\mid q}\left(1+\frac{1}{p^2}\right)-\prod_{p\mid q}\left(1-\frac{1}{p}\right)^2\right]\tag{1} $$
and by Euler's product
$$ \prod_{p\in\mathcal{P}}\left(1+\frac{1}{p^2}\right)=\frac{\zeta(2)}{\zeta(4)}=\frac{15}{\pi^2}\tag{2}$$
while
$$ \prod_{p\leq x}\left(1-\frac{1}{p}\right)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2154579",
"timestamp": "2023-03-29T00:00:00",
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Find $\frac{\sum_\limits{k=0}^{6}\csc^2\left(x+\frac{k\pi}{7}\right)}{7\csc^2(7x)}$ Find the value of
$\dfrac{\sum_\limits{k=0}^{6}\csc^2\left(x+\dfrac{k\pi}{7}\right)}{7\csc^2(7x)}$
when $x=\dfrac{\pi}{8}$.
The Hint given is: $n\cot nx=\sum_\limits{k=0}^{n-1}\cot\left(x+\dfrac{k\pi}{n}\right)$
I dont know how it come... | For future reference with this problem being tagged complex-numbers we
show how to evaluate the sum using residues. Suppose we are interested
in
$$S(n) = \sum_{k=0}^{n-1} \csc^2\left(x+\frac{k\pi}{n}\right)
= \sum_{k=0}^{n-1} \frac{2}{1-\cos\left(2x+\frac{2k\pi}{n}\right)}.$$
where we take $x$ to be a real number.
With... | {
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"url": "https://math.stackexchange.com/questions/2154639",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Finding a constant so that this is a martingale
Let $Y_{i}$ be IID with $P(Y_{i}=1)$= $\frac{1}{2}=P(Y_{i}=-1)$.
Define $K_{n}=\sum_{i=1}^{n}{Y_{i}}$.
Find the constant $r$ so that so that $$\Lambda_{n}=exp(K_{n} + rn)$$ is a Martingale.
I'm not sure that I have the right answer on this one.
Any help will be appr... | The definition of a Martingale
$$
\mathbb{E}\left[\Lambda_{n+1}|\mathcal{F}_{n}\right] = \Lambda_{n}
$$
we have
$$
\mathbb{E}\left[\mathrm{e}^{K_{n+1} + r(n+1)}|\mathcal{F}_{n}\right] = \mathbb{E}\left[\mathrm{e}^{Y_{n+1} + K_{n} + r(n+1)}|\mathcal{F}_{n}\right]
$$
we can re-write the argument as
$$
K_{n} + rn + Y_{n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2154787",
"timestamp": "2023-03-29T00:00:00",
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Proof that $a(x − y) = ax − ay$ for x and y in vector spaces. Hi i'm confused about this homework question:
"Let $V$ be a vector space over the field $F$ and let $a ∈ F$ and $x, y ∈ V$ .
Show that $a(x − y) = ax − ay$ in $V$."
What I did is below but i'm confused because I feel as though it was too simple.
So from the ... | (Responding to the suggestions in the comments.)
Recall that anything multiplied by $0$ is $0$.
If unfamiliar, you may observe that $0x + 0x = (0 + 0)x = 0x$, and use a right cancellation law on the equation $0x + 0x = 0x$ to conclude $0x = 0$.
Next, observe that $(-1)y + y = (-1)y + 1y = (-1 + 1)y = 0y = 0$.
In partic... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2154873",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proving Saalschutz Theorem I saw this in a pdf, and I'm wondering
Questions:
*
*How do you prove Saalschutz Theorem:
$$_3F_2\left[\begin{array}{c,c}-x,-y,-z\\n+1,-x-y-z-n\end{array}\right]=\dfrac {\Gamma(n+1)\Gamma(x+y+n+1)\Gamma(y+z+n+1)\Gamma(z+x+n+1)}{\Gamma(x+n+1)\Gamma(y+n+1)\Gamma(z+n+1)\Gamma(x+y+z+n+1)}\tag... | The development of the Saalschütz's identity that I know proceeds along the following path.
Start with the known identity about the sum of the product of three binomials:
$$ \bbox[lightyellow] {
\begin{gathered}
F(m,n,r,s)\quad \left| {\;0 \leqslant \text{integers}\,m,n} \right.\quad = \hfill \\
= \sum\limits_{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2155025",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 0
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Riemann-integrability of function with countably many discontinuities Enumerate the rationals in $[0,1]$ by $\{q_k\}_{k\in \mathbb{N}}$. Define for each $k$, $f_k(x) = 1$ if $q_k<x\leq1$ and $0$ if $0\leq x \leq q_k$.
I WTS $F(x) = \sum \frac{f_k(x)}{2^k}$ is Riemann integrable on $[0,1]$. I can't find an easy uniform ... | Hint:
$$\left| \frac{f_k(x)}{2^k} \right| \leqslant \frac{1}{2^k}, \\ \sum_{k=1}^\infty \frac{1}{2^k} = 1, $$
and $\sum \frac{f_k(x)}{2^k}$ is uniformly convergent by the Weierstrass test.
| {
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"url": "https://math.stackexchange.com/questions/2155181",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Analysis - Prove that there exists a smallest positive number $p$ such that $\cos(p) = 0$
Prove that there exists a smallest positive number $p$ such that $\cos(p) = 0
$.
I think I'm supposed to use either Rolle's theorem or the Mean Value Theorem but I'm not sure how, any help would be appreciated. Thanks!
| How about you restrict the domain to I=(0,pi) and show that since cos(x) is monotonic on I, it is one to one, and therefore only one value, p=pi/2, would be the solution on I since cos(a)=cos(b)=0 --> a=b, the solution p in I is unique and since any other positive solution "q" would have to be outside the interval I, p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2155270",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 3
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Product of Finite Abelian Group I need to proof for a finite abelian group $G$ that for all $x\in G$ we have $\prod_{g\in G}xg=\prod_{g\in G}g$.
I figured that using the commutative property $\prod_{g\in G}xg=x^n\prod_{g\in G}g$. Which would leave us to proof $x^n=e$, where $e$ is the identity element.
Hopefully I'm ju... | The map $g\mapsto xg$ is a permutation of $G$. hence $\prod xg$ is the same as $\prod g$, just in a different order. As $G$ is abelian, different order doesn't matter.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2155374",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is it always possible to map $n$ distinct vectors to $n$ distinct scalars? Recently while reading a paper I came across the claim that if we have $n$ distinct vectors $\vec{x}_i \in \mathbb{R}^d$ (they didn't clarify what they mean by 'distinct' but I assume it means no two vectors have all the same values) that it is ... | For any pair of distinct vectors $x_i \not= x_j \in \mathbb{R}^d$, the probability that $\langle x_i, a\rangle = \langle x_j, a\rangle$ for a uniformly random $a \in \mathbb{R}^d$ on the unit sphere is $0$.
You can see that this is true intuitively by recalling the relationship between the dot product and the angle bet... | {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "3",
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Bound on 1st Order Taylor Expansion Consider a differentiable function $f$ with the property that for any points $x,y$, we know that $\frac{|f'(x)-f'(y)|}{|x-y|} \leq C.$ How do I show that the following inequality holds for this function: $$f(x) - f(y) - f'(y)(x-y) - \frac{C}{2}(x-y)^2 \leq 0$$
Note the similarities t... | I'll write this out here for clarity:
From the Lagrange form of Taylor's theorem, we have that
$$f(x) = f(y) + f'(y) (x-y) + \frac{f''(\xi)}{2}(x-y)^2$$
for some $\xi \in (x, y)$. So if we can show that $f''(\xi) \leq C$ for any $\xi \in (x, y)$, we are done.
Using the inequality given in the question, since it holds ... | {
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"timestamp": "2023-03-29T00:00:00",
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What is the sum of the solutions to $6x^3+7x^2-x-2=0$ What is the easy way to solve the problem?
The sum of the solutions to $6x^3+7x^2-x-2=0$ is:
$$A) \ \frac{1}{6}$$ $$B) \ \frac{1}{3}$$ $$C) \ \frac{-7}{6}$$ $$D) -2$$ $$E) \text{ none of above}$$
| For a cubic equation of the form $ax^3 + bx^2 + cx +d$, the sum of all the roots is given by $\frac{-b}{a}$. Here, $b=7,a=6$ so sum of the roots $= \frac{-7}{6}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2155713",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Sample space and total possible events Taken from my book:
How many events are there associated with the roll of one die? Solution: Each event corresponds to a subset of {1,2,3,4,5,6}. there are $2^6$ subsets, so there are $2^6$ possible events.
Now my question is: why $2^6$ events? If you roll only one dice shouldn't ... | You have the terms "event" and "outcome" confused.
The outcomes of the sample space are: $1, 2, 3, 4, 5,$ and $6$. These are the elements of the sample space: $\{1, 2, 3, 4, 5,6\}$ .
An event is a set of outcomes. Such as the event of rolling an even number: $\{2,4,6\}$, the event of rolling a number greater than... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2155961",
"timestamp": "2023-03-29T00:00:00",
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Prove the equivalence of norms on the Hardy space $H^2(\mathbb{D})$. Let $H^2(\mathbb{D})$ be the space of all functions $f$ holomorphic on the open unit disk $\mathbb{D}$ such that the Hardy norm, given below, is finite:
$$||f||_H^2 = \sup_{0<r<1}\frac{1}{2\pi}\int_0^{2\pi} |f(re^{i\theta})|^2 \ \mathrm{d}\theta.$$
I ... | The key tools here are the closed graph theorem, the open mapping theorem (for Banach spaces), and the identity principle for holomorphic functions.
So suppose $f_m \to f$ in $(H^2,\|\,\|_H)$ and $f_m \to g$ in $(H^2,\|\,\|).$ Because we know point evaluation is continuous on the first space, we have
$$f_m(1/(n+1)) \to... | {
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"timestamp": "2023-03-29T00:00:00",
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Distribution of the squared sum of a categorical distribution Given $k\in\mathbb{N}^+$ and $P = \{p : p \in [0, 1]^k \wedge \sum_{i=1}^k p_i = 1 \}$, we choose a random element $q$ of $P$. What is the probability distribution of $S = \sum_{i=1}^k q_i^2$?
I have made some plots with synthetic examples and all I have fig... | A common distribution on the simplex is the Dirichlet Distribution. See that your example we set all $\alpha_i=1$ to get $\mathbb{E}[p_i^2]=\frac{1}{k^2}+\frac{k-1}{k^2(k+1)}=\frac{2}{k(k+1)}$, and so $\mathbb{E}[S]=\frac{2}{k+1}$. As far as distributions go the $p_i$ are beta which means their squares don't have an el... | {
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"url": "https://math.stackexchange.com/questions/2156228",
"timestamp": "2023-03-29T00:00:00",
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Minimize $P=5\left(x^2+y^2\right)+2z^2$ For $\left(x+y\right)\left(x+z\right)\left(y+z\right)=144$, minimize $$P=5\left(x^2+y^2\right)+2z^2$$
I have no idea. Can you make a few suggestions?
| HINT: prove that $$P=5(x^2+y^2)+2z^2\geq 72$$ and the equal sign will atained at $$(x,y,z)=(2,2,4)$$
| {
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How to minimize a sum of absolute values using linear programming? I am having trouble understanding the logic behind optimization of cost function of the form $$\min (|x| + |y| + |z|) \,$$ subject to constraints $$Ax \le b \qquad Cx = d $$ such as $$ x + y \le 1 \qquad 2x + z = 3.$$
I have seen methods involving repre... | On that page, they are solving another problem. They are trying to make the constraint |x| < b, so it's true that they can split that inequation in those two. In your case, you can't, because you need it to be either x, or -x, not any value in between.
What you can do, is using a bivalent variable. Those are also calle... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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A Tic-Tac-Toe Variant Let's imagine a game like tic-tac-toe, but you have to have 3 of the same sign in the same line or colon... So the diagonals don't count!
Then I have to prove that the second player to play can always be sure not to lose... But how do I do that?
I am searching a simple argument, not just like cons... | Suppose X plays first. The only thing O needs to do to avoid losing is to manage to have one mark in each row and one mark in each column. It's easy to find a strategy that can achieve that.
For example, place the first two Os in the same column and the same row as X's first play. Before O's next move, the board looks ... | {
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How to find an equation for this sine-ish wave I'm programming a game. Part of the game involves a spinning rectangle, so I'd like to keep track of two points on that rectangle. The center, and the bottom right corner of it.
I want to draw a sine wave of the relative vertical distance between, the center and the corne... | Your corner has the initial coordinates $$\pmatrix{x_0\\y_0} = \pmatrix{32\\-16}.$$ In order to get the new coordinates, you have to apply the rotation, which can be expressed in terms of a matrix-vector multiplication:
$$
\pmatrix{x\\y} = \pmatrix{\cos\varphi & -\sin\varphi \\ \sin\varphi & \cos\varphi} \pmatrix{x_0\\... | {
"language": "en",
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How to find the Laplace Transform of $t^2sin(t)$ Using the rule: $$\mathcal{L}(t^nf(t))=(-1)^n\frac{d^n}{ds^n}F(s)$$
where in this case
$$f(t)=\sin(t),\,\,\,\,\,\,\,\,\,\mathcal{L}(\sin(t))=F(s)=\frac1{s^2+1},\,\,\,\,\,\,\,\,\,\,n=2.$$
Find the 2nd derivative of F(s):
$$\frac{d^2}{ds^2}\Big(\frac... | $L\{tf(t)\} = \int_0^{\infty} tf(t) e^{-st} dt$
if we integrate both sides with respect to $s.$
$\int L\{tf(t)\} ds = -\int_0^{\infty} f(t) e^{-st} dt = -L\{f(t)\}$
And then we can differentiate both sides (with respect to $s$).
$L\{tf(t)\} = -\frac {d}{ds} L\{f(t)\}$
$L\{t^2\sin t\}=\frac{d^2}{ds^2} \frac {1}{s^2+1}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2156907",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Probability problem with $n$ keys and $k$ locks Let's say we have $n$ keys and $k$ locks on a door, with $n \ge k$. The $n$ keys are different, i.e. only $1$ key goes into a specific lock.
Also every lock has a key that goes to it (but some of the keys may be "duds").
What is the probability of unlockin gthe door on th... | Note: I think the question may be defining "attempt" as trying the key on a single lock. In this answer, I took "attempt" to be trying one key (on every lock).
The question is: if there are $n$ keys and $k$ locks, and a random ordering of the $n$ keys, what is the probability that the locks are all unlocked in $m$ step... | {
"language": "en",
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A self inverse function $f(x)=\frac{ax+1}{x-b}$
If $$f(x)=\frac{ax+1}{x-b} \forall x \in\mathbb{R}-b,ab\neq1,a\neq1$$ is a self inverse function such that $$\frac{f(4)}{4}=\frac{f(12)}{12}={f\left(\frac{1+b}{1-a}\right)}$$The question is to find out $a$ and $b$
For a self inverse function $f(f(x))=x$.So I tried to p... | Write this as
$$\begin{pmatrix}
a && 1 \\
1 && -b
\end{pmatrix}\begin{pmatrix}x \\ 1\end{pmatrix}$$
Then this is self inverse iff
$$\begin{pmatrix}
a && 1 \\
1 && -b
\end{pmatrix}^2= \lambda I_2$$
i.e.
$$\begin{pmatrix}
a^2+1 && a-b \\
a-b && 1+b^2
\end{pmatrix}= \lambda I_2$$
So $a=b$.
The transformation is projecti... | {
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"timestamp": "2023-03-29T00:00:00",
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The deep reason why $\int \frac{1}{x}\operatorname{d}x$ is a transcendental function ($\log$) In general, the indefinite integral of $x^n$ has power $n+1$. This is the standard power rule. Why does it "break" for $n=-1$? In other words, the derivative rule $$\frac{d}{dx} x^{n} = nx^{n-1}$$ fails to hold for $n=0$.
... | The so-called "deep reason" is not deep at all. The term $\displaystyle \log(x)$ is simply the constant term in the expansion of $\displaystyle \frac{x^{n+1}}{n+1}$ around $n=-1$. To see this, we simply write
$$\begin{align}\frac{x^{n+1}}{n+1}&=\frac{e^{(n+1)\log(x)}}{n+1}\\\\&=\frac{1}{n+1}\sum_{k=0}^\infty \frac{(n... | {
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Find the limit of the following log terms
I thouht about it alot but could not able to getany idea .
Can anybody provide me a hint
| Hint:
$$\lim_{x \to 0} \frac{f(3+x) - f(3-x)}{x} = \lim_{x \to 0} \frac{f(3+x) - f(3)}{x} + \lim_{x \to 0} \frac{f(3-x) - f(3)}{-x} = f'(3) + f'(3)$$
| {
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If $a^2+b^2+c^2+d^2+e^2=5$ so $\sum\limits_{cyc}\frac{1}{7-2a}\leq1$.
Let $a$, $b$, $c$, $d$ and $d$ be non-negative numbers such that $a^2+b^2+c^2+d^2+e^2=5$. Prove that:
$$\frac{1}{7-2a}+\frac{1}{7-2b}+\frac{1}{7-2c}+\frac{1}{7-2d}+\frac{1}{7-2e}\leq1$$
The equality occurs also for $a=2$ and $b=c=d=e=\frac{1}{2}$... | We must to prove that $f(a_1)+f(a_2)+f(a_3)+f(a_4)+f(a_5) \le 1$ ,
for non-negative $a_1+a_2+a_3+a_4+a_5=5$
$$f(x)=\dfrac{1}{7-2\sqrt{x}}$$
Since $f''(x)=\dfrac{(7-6\sqrt{x})}{2(2x-7\sqrt{x})^3}$ , we only need to consider the inequality in case $0< a_1=a_2=a_3=a_4=t^2 \le 1 \ , \ a_5=5-4t^2$
Clearly
$g(t)=\dfrac{4}{7... | {
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Why does $[f'(a)(a+\Delta x-a)+b]-[f'(a)(a-a)+b]=f'(a)\Delta x$ Why does $[f'(a)(a+\Delta x-a)+b]-[f'(a)(a-a)+b]=f'(a)\Delta x$? If I try to calculate this I get $f'(a)(\Delta x-1) \neq f'(a)\Delta x$.
| Some of the terms cancel. Look:
$$\begin{align}
[f'(a)(\color{red}{a}+\Delta x\color{red}{-a})+b]-[f'(a)\color{blue}{(a-a)}+b]&= [f'(a)(\Delta x)+b]-[f'(a)\cdot \color{blue}{0}+b] \\
&= f'(a)(\Delta x) \color{green}{+b-b} \\
&= f'(a)(\Delta x).\end{align} $$
| {
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Gradient and transpose I have simple question, when taking the Jacobian of a linear vector function, do you have to transpose the matrix $A$?
$b\in\mathbb{R}^{n},x\in\mathbb{R}^{m}\\J(Ax+b) = A^T, A\in\mathbb{R}^{n\times m}$
| $J_x(f)$ is the, in case of existence, uniquely determined matrix with the property
$$f(x+h)=f(x)+J_x(f) h+\mathrm o(\Vert h\Vert).$$
When plugging in $f(x)=Ax+b$, you are looking for a $J$ with
$$ A(x+h)+b=Ax + b + J_x(f)h+\mathrm o(\Vert h\Vert).$$
Since the left side is equal to $Ax + Ah + b$ (because of linearity... | {
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Number of conjugacy classes are there over $\mathbb{F}_p$ I am struggling with the next exercise of my HW:
How many conjugacy classes are in $GL_3(\mathbb{F}_p)$? And how many in $SL_2(\mathbb{F}_p)$?
It's on the topic of Frobenius normal form of finitely generated modules over $\mathbb{F}_p$.
I'd appreciate any idea.
| For the general linear group I suggest that you count the (irreducible) linear polynomials with non-zero constant term, the irreducible quadratics, and finally the irreducible cubics. You then get a conjugacy class for each cubic, $C(f(X)$; a conjugacy class for each pair (linear, irreducible quadratic), $C(X-\alpha)\o... | {
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Can a group always be written as a Zappa-Szep product of its Sylow subgroups? In particular, can a simple group (so that we don't have any proper normal subgroups) have an exact factorization, i.e. do there always exist $H, K<G$ with $H\cap K = \{1_G\}$ and $|H|\cdot|K|=|G|$?
| To answer the question in the title, the group ${\rm PSU}(3,3)$ of order $6048 = 2^5.3^3.7$ cannot be wriiten as a product $P_1P_2P_3$ of $3$ Sylow subgroups in any order. That is proved in a paper I wrote with Petey Rowley a while ago:
D. Holt and P. Rowley, On products of Sylow subgroups in finite groups,
Arch. Math... | {
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Which of the following numbers is greater? Which of the following numbers is greater?
Without using a calculator and logarithm.
$$7^{55} ,5^{72}$$
My try
$$A=\frac{7^{55} }{5^{55}×5^{17}}=\frac{ 7^{55}}{5^{55}}×\frac{1}{5^{17}}= \left(\frac{7}{5}\right)^{55} \left(\frac{1}{5}\right)^{17}$$
What now?
| Note: $7^2<2\cdot 5^2$ and $5>2^2$
$7^{55}<7\cdot 5^{54}\cdot 2^{27}<5^{55}\cdot 2^{28}<5^{69}<5^{72}$ as required
With an extra jink into factors of $3$, we can show $7^{55}<5^{67}$
Extra notes: $3^3>5^2$ and $5^5>3\cdot2^{10}$
$7^{55}<7\cdot 5^{54}\cdot 2^{27}<5^{54}\cdot 2^{30}<5^{52}\cdot 2^{30}\cdot 3^{3}<5^{67}... | {
"language": "en",
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Studying properties of an integral $f:\mathbb R \to \mathbb R, \: f(x) =\int _0^x\:e^{-t^3}\left(t^3+t^2-t-1\right)dt$
Firstly, I have to find $min(f(x))$. The possible answers are:
A) $f(0)$
B) $f(1)$
C) $f(-1)$
D) $f(3)$
E) $f$ tends to $- \infty$
In order to do that, I have rewritten the function like this
$f\left(x... | Apply Newton-Leibnitz theorem. Then note that max or min is achieved at $0$ for derivative . Now differentiate the derivative to get second derivative by product rule. See for which of the root of first derivative the value of second derivative is positive(because thats the condition for minima. Hope you know why)
| {
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Distribution Function Evaluated With Inequality Let's say you have a distribution function F. According to Intro to Mathematical Statistics (Hogg), the following property holds:
Pr($a < X \leq b$, $c < Y \leq d$) = $F(b,d) - F(b,c) - F(a,d) + F(a,c)$
What is the intuition behind this property, and how can you prove th... | It is just the Principle of Inclusion and Exclusion.
Consider the quadrants on a Venn Diagram : $~{\bbox[blue]{\Box}~\bbox[white]{\Box}\\\bbox[purple]{\Box}~\bbox[red]{\Box}}$
$$\def\P{\operatorname{\sf P}} \begin{align} \P((X,Y)\in(a;b]{\times}(c;d]) & =\P(\Box) \\[1ex] & = \P\left({\bbox[blue]{\Box}~\bbox[white]{\Box... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Asymptotic estimate of the form $k=C\log\log n$. Suppose that we have
$$
n=k^{\frac{1}{\beta}k^22^{k+12}\log\frac 1{\alpha}}
$$
where $\alpha,\beta\in(0,1)$. It is claimed that
$$
k\sim C\log\log n
$$
for some constant $C=C(\alpha,\beta)$. How do one deduce such an estimate?
I am totally lost here, any help is very muc... | I assume logarithms in base $2$. This would not change anything anyway, besides constants.
First, note that
$$
n=k^{\frac{1}{\beta}k^22^{k+12}\log\frac{1}{\alpha}}
= 2^{\frac{1}{\beta}k^22^{k+12}\log\frac{1}{\alpha} \log k}
= 2^{\frac{2^{12}\log\frac{1}{\alpha}}{\beta}k^22 ^{k} \log k}
$$
so
$$
\log n = \frac{2^{12}\lo... | {
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Reference request regarding Mackey topology I would like to ask for some request on the relationship between Mackey topology and the weak topology. Recently, I have read quite some statements where a property in weak topology (such as compactness) implies that in Mackey topology. The basic connection I have now is that... | The basic reference is Topological Vector Space, the second is Bourbaki's classic. Other books on Topological Spaces also can be mentioned. However, I want to mention another direction which might interest you.
The Banach Lattice Theory (or more general form), makes it very easy to find Mackey topology. The most import... | {
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What is the number of rooted planar decreasing trees on n vertices? By planar, I mean embedded in the plane (I think sometimes the terms, ordered and plane, are used interchangeably with planar).
Assume the vertices are labeled 1,2,...,n. By decreasing, I mean the labels on the vertices along a path from the root to a... | Let's call these ordered decreasing trees as the term planar is also
used for the cyclic group acting at the root. Now we have the
following recursive combinatorial construction. To assemble one of
these we need a root node, which receives the label $n$ and an ordered
sequence of subtrees, each of some s... | {
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How do you know what $\sum_{n=a}^{\infty}p^n$ converges to if you know $\sum_{n=0}^{\infty}p^n = \frac{1}{1-p}$? How do you know what $\sum_{n=a}^{\infty}p^n$ if you know $\sum_{n=0}^{\infty}p^n = \frac{1}{1-p}$ ?
Apparently $\sum_{n=a}^{\infty}p^n$ = $\frac{p^a}{1-p}$, but how can you derive this?
| Because of the high-school identity:
$$1-p^n=(1-p)(1+p+p^2+\dots+p^{n-1}),$$
rewritten as
$$\frac1{1-p}=1+p+p^2+\dots+p^{n-1}+\frac{p^n}{1-p}, $$
which shows that
$$\Biggl\lvert\frac1{1-p}-(1+p+p^2+\dots+p^{n-1})\Biggr\rvert=\frac{\lvert p^n\rvert}{\rvert 1-p\rvert},$$
and the latter fraction tends to $0$ if $\lvert p... | {
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Proving $ 2n^{n-3} = \sum\limits_{i=1}^{n-1}\binom{n-2}{i-1}i^{\ i-2}(n-i)^{\ n-i-2} $ My question is how to show$$ 2n^{n-3} = \sum_{i=1}^{n-1}\binom{n-2}{i-1}i^{\ i-2}(n-i)^{\ n-i-2} $$
I got to this result through a problem of counting labeled trees, but when I try to approach actually computing the sum, I've had no ... | Start by re-writing as follows:
$$2n^{n-3} = \sum_{k=0}^{n-2}
{n-2\choose k} (k+1)^{k-1} (n-1-k)^{n-3-k}$$
so that we seek to verify that
$$\bbox[5px,border:2px solid #00A000]{
Q_n = \sum_{k=0}^n {n\choose k}
(k+1)^{k-1} (n+1-k)^{n-1-k} = 2(n+2)^{n-1}.}$$
Concerning the exponential generating function for this quantit... | {
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How smooth is the generalized power mean function? Consider the function of power mean of $1$ and $e$ (so that the computation is simpler). That is, define $\displaystyle f(x)=(\frac{1+e^x}{2})^{1/x}$ when $x\neq 0$ and $f(0)=\sqrt{e}$.
It's not difficult to use L'Hopital rule with std. exponential trick to verify the ... | I will stick to $x\in \mathbb R.$ We can write
$$\tag 1 f(x) = \exp [(1/x)\ln ((e^x+1)/2)]$$
for $x\ne 0.$ Now $\ln ((e^x+1)/2)$ is real analytic on $\mathbb R.$ Why? Because $(e^x+1)/2$ is real analytic and positive on $\mathbb R,$ $\ln x$ is real analytic on $(0,\infty),$ and compositions of real analytic functions a... | {
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"timestamp": "2023-03-29T00:00:00",
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A question on roots of a Quadratic. If roots of the equation ${ax}^2+bx+c=0$ are of the form $$\frac{\alpha}{\alpha-1},\frac{\alpha+1}{\alpha}$$
Then the value of $$({a+b+c})^2$$
I have no clue how to approach this one, any help is appreciated!
| From the form:
$\frac{1}{x} + x = 2$ => $x^2 + x -2 = 0$, and first equation is $kx^2 + kx - 2k = 0$.
So, $(a+b+c)^2=0$
| {
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The derivative of $(\log_2 n)^5$? The derivative of $ (\log_2 n)^5$? (log base 2)
Hey everyone,
I am not sure how to go about this question because I am not sure what to do with the power $5$ ( or any other power ) in the log? Should I try to convert it to a natural log ? How ? Any help ?
(Note: this is not a HW probl... | Note that I'm implicitly assuming that $n \in \mathbb R$. First, we observe that
$$\log_2 n=\log(n)/\log(2)$$
where $\log$ is the natural logarithm. Hence,
$$(\log_2n)^5=\left(\frac{\log n}{\log 2}\right)^5=\frac{1}{\log(2)^5} \cdot \log(n)^5.$$
To find the derivative, we use the chain rule: $g(f(x))^\prime=g^\prime(f... | {
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"timestamp": "2023-03-29T00:00:00",
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Distribution of two random variable generating schemes Goal: Generate five numbers from 0 to 1 with sum 1.
Method 1: Generate four numbers in range $(0,1)$ (by uniform distribution) to be the cuts of the interval, i.e. say the four random numbers generated is $a_1<a_2<a_3<a_4$, then the five random numbers are $a_1, a_... | Let's play this game in a simpler case. Let $X_1$ and $X_2$ be two independent and $U(0,1)$ distributed random variables.
FOR METHOD 1
Generate random variables $A_1$ and $A_2$ such that $A_1+A_2=1$ the following way:
$$A_1=\begin{cases}X_1&\text{ if }& X_1<X_2\\
X_2&\text{ if }& X_2\le X_1\end{cases} \text{ and let }... | {
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Which of the following $u=\sqrt{c+1} - \sqrt{c}$ and $v=\sqrt{c} - \sqrt{c-1}$ is greater? If
$$u=\sqrt{c+1} - \sqrt{c}$$
and
$$v=\sqrt{c} - \sqrt{c-1}$$
then, which among $u$ and $v$ is greater?
Please help without substituting values.
| Note that $\sqrt x$ is concave function, so $$2\sqrt x \gt \sqrt {x-1} + \sqrt {x+1}$$
| {
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Requesting suggestion on geometry and calculus books i am an engineering student , i had little interest in mathematics but my interest in mathematics inadvertently( exactly because i am loving that) rises to extreme level, although i have little knowledge (subject-syllabus oriented) on mathematics, i am eagerly tryin... | O'Neil's differential geometry is probably a great place to learn both calculus and some elementary differential geometry at the same time! It has been posted to the internet archive. It isn't so easy to find a physical copy of this book, unless you're at a large university. https://archive.org/details/ElementaryDiffer... | {
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How to solve this calculus problem This question was on my quiz for Calc AB, everyone was confused on how to solve it and I did some random substitution but was still getting nowhere. My teacher said it's possible to do it but not possible with the knowledge we have currently and she's going to throw out the problem. T... | Multiply by $e^{-x}$:
$$y'e^{-x}-ye^{-x}=-3xe^{-x}$$
Integrating we obtain: $$ye^{-x}=3(x+1)e^{-x}+C$$
$$y=3x+3+Ce^{-x}$$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Evaluate: $\int_{0}^{\pi}\frac{\cos 2017x}{5-4\cos x}dx$ Evaluate:
$\int\limits_{0}^{\pi}\dfrac{\cos 2017x}{5-4\cos x}~dx$
I thought of using some series but could not get it
| $\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{\displaystyle{#1}}\,}
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Solve $\lim_{x \rightarrow e} \frac{e-x}{\ln x -1}$ without using L'Hopital's rule I tried:
$$\lim_{x \rightarrow e} \frac{e-x}{\ln x -1} = \frac{e-x}{\ln(x)-\ln(e)} = \frac{e-x}{\ln(\frac{x}{e})} = ???$$
What do I do next? I think I could use $\lim \frac{\log_a x}{x} = 0, a>1$, but I'm not sure how.
| To avoid the derivative (since this is effectively L'Hospital's rule), notice that
$$\lim_{x \rightarrow e} \frac{e-x}{\ln x -1} = -e\lim_{x \rightarrow e} \frac{\frac{x}{e} - 1}{\ln \frac{x}{e} } = -e \lim_{y \to 1} \frac{y-1}{\ln y}.$$
Using the well-known inequality $(y-1)/y \leqslant \ln y \leqslant y - 1$ it follo... | {
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Find the value of $\arctan(1/3)$ How can I calculate $\arctan\left({1\over 3}\right)$ in terms of $\pi$ ? I know that $\tan^2(\frac{\pi}{6})= {1\over3}$ but don't know if that helps in any way.
| The numerical computation of $\arctan\frac{1}{3}$ is pretty simple from the Maclaurin series of $\arctan$:
$$\arctan\frac{1}{3}=\sum_{n\geq 0}\frac{(-1)^n}{(2n+1)3^{2n+1}} \tag{1}$$
To get $\arctan\frac{1}{3}\approx 0.321751$ we just need to consider the partial sum up to $n=4$.
By multiplying this constant by $\frac{... | {
"language": "en",
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What will happen to the roots of $ax^2 + bx + c = 0$ if the $a \to 0$? Exercise:
What will happen to the roots of the quadratic equation
$$ax^2 + bx + c = 0$$
if the coefficient $a$ approaches zero while the coefficients $b$ and $c$ are constant, and $b \neq 0$?
Attempt:
$\lim\limits_{a \to 0}{(ax^2 + bx + c)} =... | If $b=0$, then the roots are (if real), $\pm\sqrt{-c/a}$. If $c\ne0$, both roots tend to infinity (positive and negative).
If $b\ne0$, it's not restrictive to assume $b>0$ (otherwise multiply by $-1$).
If $c=0$, the roots are $0$ and $-b/a$, the latter tending to infinity ($\infty$ if $a$ approaches $0$ from the negat... | {
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Solving the homogeneous heat equation using the method of fundamental solutions I'm trying to solve the PDE $u_t=Du_{xx}$ using the method of fundamental solutions. I've used the ansatz $u(x,t)=t^{\alpha/2}f(x/\sqrt{t})$ and this has given me the following ODE: $$\frac{\alpha}{2}f(\phi)-\frac{1}{2}\phi f'(\phi)=Df''(\p... | Let's start from the top. Make the ansatz
$$
u(x,t) = t^\alpha f\left(\frac{x}{2\sqrt{Dt}}\right)
$$
Then after some massaging of the differential equation and substituting $\phi = x/(2\sqrt{Dt})$, you end up with
$$
f''(\phi) + 2\phi f'(\phi) - 4\alpha f(\phi) = 0
$$
The form of the differential operator suggests the ... | {
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The derivative of $x^0$ For some reason I have not been able to find a straight answer to this.
We know that $\frac{d}{dx}x^n=nx^{n-1}$
And this is true for $n=-1$ and $n=1$ $\implies$ $\frac{d}{dx}x^{-1}=-1x^{-2}$ and $\frac{d}{dx}x^1=1$
We also know that $\frac{d}{dx}C=0$ where $C$ is a constant.
Suppose that $f(x)=x... | You're incorrectly applying the power rule. We have that $\frac{d}{dx}x^n = nx^{n-1}$. For $x^0$, $n = 0$. So $\frac{d}{dx}x^0 = 0x^{-1} = 0$.
$x^1$ does not simplify to a constant. $x^1$ is just $x$, which is anything but constant.
| {
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Proof by contradiction: There are infinitely many primes I need some help with a proof. I just need to be pointed in the right direction, because I've been looking at this for ages and it's not clicking.
I need to prove that there are infinitely many prime numbers, by contradiction. The original statement is:
For all ... | Case 1, as you already said, is trivial.
Case 2 is a little more tricky. Suppose that n!-1 is composite. Then it must be divisible by at least two primes, as you have already stated. But since n! is divisible by all numbers less than n, consider- what numbers less than n could go into n!-1?
| {
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Calculate using Euler integrals $\int \limits_0^{+\infty} \frac{1}{x^{n+1}} e^{-\frac{\alpha}{2x^2}}dx$ Find existence region and express the function in terms of Euler's integrals
$$\int \limits_0^{+\infty} \frac{1}{x^{n+1}} e^{-\frac{\alpha}{2x^2}}dx, \alpha > 0, n \in N$$
I have tried $t = \frac{1}{x}, dx = - \fra... | Well, we have that:
$$\mathscr{I}\left(\text{a},\text{n},x\right):=\int\frac{\exp\left(-\frac{\text{a}}{2x^2}\right)}{x^{1+\text{n}}}\space\text{d}x\tag1$$
Substitute:
$$\text{u}=\frac{\text{a}^\frac{\text{n}}{2}}{2^\frac{\text{n}}{2}\cdot x^\text{n}}\tag2$$
So, we get that:
$$\mathscr{I}\left(\text{a},\text{n},x\right... | {
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"timestamp": "2023-03-29T00:00:00",
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Proof for Property of Complex Numbers Is the inequality $\lvert z_1 + z_2 \rvert \ge \lvert z_1 \rvert - \lvert z_2 \rvert$ incorrect, where $z_1$ and $z_2$ are any two complex numbers? I need an example to prove that it is. And in case it is correct, can you please give the proof? Thanks for any help.
| This inequality is always true. It follows easily by the triangle inequality.
Start with $$\vert z_1\vert\le\bigl\vert z_1-(-z_2)\bigr\vert+\left\vert-z_2\right\vert\;.$$
| {
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Prove that every translation or rotation of $\Bbb E^2$ is a composite of rotations about $P$ and $Q$. Prove that every translation or rotation of $\Bbb E^2$ is a composite of rotations about $P$ and $Q$.
| Lemma 1. For $d$ with $0\le d\le 2\operatorname{dist}(P,Q)$, there exists a translation by $d$ in a suitable direction, that can be achieved.
Proof. Rotating around $P$ by $\alpha$ and around $Q$ by $-\alpha$ produces a translation where the distance (and direction) depends continuously on $\alpha$. For $\alpha=0$, the... | {
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"source": "stackexchange",
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If the Gateaux derivative is not linear does this mean the Frechet derivative doesn't exist? It seems to be true as if the frechet exists then it is the same as the gateaux which would then be nonlinear (contradiction) but this seems useful yet I can't really find it anywhere and the lecturer hasn't mentioned it. Thank... | I don't when if you are working in Euclidean spaces or normed spaces, but
anyway, Frechet differentiability at a point $x_{0}$ for a function
$f:E\rightarrow Y$, where $E\subseteq X$, means
$$
\lim_{x\rightarrow x_{0}}\frac{\Vert f(x)-f(x_{0})-L(x-x_{0})||_{Y}}{\Vert
x-x_{0}||_{X}}=0
$$
where $L:X\rightarrow Y$ is line... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to map a square to a triangle? I need the opposite direction, I was not quite seeing it. I then decided to try and find a transformation from square to triangle in a hope to invert it.
I am trying to find an invertible transformation between the unit square $x,y \in[0,1] \times [0,1]$ and the triangle $T$ bounded ... | From this page !!!.${}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2160984",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Physical meaning of one dimensional heat equation. Consider heat conduction in a $1$-D material. The temperature distribution $u$ after sufficiently long time can be modeled by $$-(a(x)u_x)_x = f(x), \qquad x \in [0,1]$$ where $a$ is heat conductivity, $f$ denotes internal heat sources.
Now i want to understand how t... | You can start from the derivation of the general heat equation in one dimension. For example, you can look at the https://en.wikipedia.org/wiki/Heat_equation. You just need to account for the fact that the heat conductivity is position dependent. You should get:
$$\partial_tu-\partial_x(a(x)\partial_xu)=f(x)$$
Now all ... | {
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"timestamp": "2023-03-29T00:00:00",
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The natural topology on a $C^k$-manifold Lang- Differential Manifolds p.21
First Question. As it is written above, the author says it is trivial to prove that one can give $X$ a topology in a unique way such that $U_i$ is open, and the $\phi_i$ are topological embeddings, but how?
Second Question. Let $\mathscr{A},\ma... | The topology you want is defined as follows: $V \subset M$ is open iff $\varphi_i(V \cap U_i)$ is open in $\mathbf E_i$ for all $i$.
With this definition, the answers to the next two questions are clearly "yes".
My favourite textbook on this sort of stuff is Berger and Gostiaux - see 2.2.6, 2.2.7 and 2.2.8, which provi... | {
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How to find linear programming constraints given a 2 player zero sum matrix Given a problem like the following:
A startup want to build talking washing machines spending the least possible. There are three ways of building them: manually, semi-automatically and automatically. The manual production demands 1 minute of ... | Your setup is perhaps confusing because it's not clear what $a,b,z$ represent, and there's nothing relating them to each other or constraining them. As it stands, you could pick any feasible solution (e.g., $(x_1,x_2,a,b) = (1,0,1,-1)$), and then an arbitrarily low value for $z$. This shows that the problem is unbounde... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Factors of integers of the form $p-2^\lambda n$. Here $p$ is an odd prime, $n$ is uniform on $[0, 2^\lambda]$, and $\lambda$ is a constant. We define distribution $\mathcal{D}$ by:
$$x \xleftarrow{\$} p-2^\lambda n$$
Assume $p \approx 2^{4\lambda}$, $\lambda \in \{128, 256\}$, and $0 \leq k \leq \log_2 \lambda$. Do a n... | So I've been thinking this all day, read an article on wired. https://www.wired.com/2014/12/mathematicians-make-major-discovery-prime-numbers/
If you consider the prime number theorem, $ \pi(n)~ N/logN $ it says that the distribution of primes increases with the $log(N)$.
Because you have a static distribution over y... | {
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"source": "stackexchange",
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Sequence Limit: $\sin(n^3)$ How can I prove that this sequence does not converge, using the definition?
$$W_n = \sin(n^3)$$
For $n \in \mathbb{N}$. I tried to do a proof by reduction to the absurd but without result.
| Reference: https://terrytao.wordpress.com/2010/03/28/254b-notes-1-equidistribution-of-polynomial-sequences-in-torii/
As a special case of Corollary 6, we have
Let $P(n)=a_s n^s + \cdots + a_0$ be a polynomial with real coefficients. If $a_s$ is irrational, then $P(n)$ mod $1$ is equidistributed.
With $P(n)=\frac1{2\... | {
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"url": "https://math.stackexchange.com/questions/2161491",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Solving or approximating infinitely nested integral Let $f$ given by
$$f(x) = g(x) + \int_0^x\left(g(x_1) + \int_0^{x_1} \left( g(x_2) + \int_0^{x_2} \ldots d_{x_n} \ldots \right) d_{x_2} \right) d_{x_1}$$
where $n \rightarrow \infty$ and $g(x)$ is strictly decreasing in $x$.
How can such an integral be solved or appro... | If $$h_n(x):=g(x) + \int_0^x\left(g(x_1) + \int_0^{x_1} \left( g(x_2) + \int_0^{x_2} \ldots d_{x_n} \ldots \right) d_{x_2} \right) d_{x_1} $$ converges to a limit function $h$, then the self similar nature of this expression implies that
$$g(x)+\int_0^x h(t) \mathrm{d}t=h(x). $$
Differentiating gives
$$g'(x)+h(x)=h'(x)... | {
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Prove that $(\sum_{i=1}^n i)^2$ = $\sum_{i=1}^n i^3$ by induction Prove that: $(\sum_{i=1}^n i)^2$ = $\sum_{i=1}^n i^3$
I can use the fact that $\sum_{i=1}^n i$ = n(n+1)/2 after the inductive hypothesis is invoked.
I'm not sure where to start, I would usually break down one side but there isn't usually two sums, so I'm... | If $n = 1$, the hypothesis is true.
Assume that the hypothesis is true for $n =k$ i.e.
$(\sum_{i=1}^{k} i)^2 = \sum_{i=1}^{k} i^3$.
Now try to prove that the hypothesis is true if $n =k+1$.
| {
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what is the definition of local base Hi guys I am facing problem while understanding the definition of a local base.
Can anybody tell me the difference/relation between the following two definitions?
Def $1$: "Let $p$ be any arbitrary point in a topological space $X$. Then a class $B_p$ of OPEN SUBSETS of $X$ is called... | If we have an "open local base", it is of course a "neighbourhood local base" as well.
This is because $N$ is a neighbourhood of $x \in N$ iff there exists an open set $O$ such that $x \in O \subseteq N$. So by definition (take the set itself every time), an open set is a neighbourhood for each of its points (and this... | {
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Primality test for numbers of the form $2p+1$ Is this proof correct ?
Claim
Let $p \equiv 5 \pmod 6$ be prime , then $2p+1$ is prime iff $2p+1 \mid 3^p-1$ .
Proof
Suppose $q=2p+1$ is prime. $q \equiv 11 \pmod{12}$ so $3$ is quadratic residue module $q$
and it follows that there is an integer $n$ such that $n^2 \equ... | Your proof is pretty much correct (the other answers have already mentioned how you could improve the last line), so let me give a somewhat easier proof of the second part. It only uses the Fermat-Euler theorem and some properties of the $\phi$ function.
Let $d$ be the smallest positive integer with:
$$3^d\equiv 1\pmod... | {
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What meaning can be given to a permutation to the power of another permutation? Say you have 2 permutations, f and g, how would one calculate $g^f$? Also can you multiply these permutations and how?
| Usually in group theory $g^f$ is used notation for conjugate of $g$ by $f$, i.e. $g^f := f^{-1}gf$. You can calculate this product in the following manner. If $g = (a_1a_2\cdots a_n)$, then $g^f = f^{-1}gf = (f^{-1}(a_1)f^{-1}(a_2)\cdots f^{-1}(a_n)$).
On the otherside multiplication of permutations is composition of t... | {
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Show that $2^a|(x-1)$ iff $2^a|(y-1)$. Suppose that $2^n + 1 =xy$, where $x$ and $y$ are integers > $1$ and $n>0$.
Show that $2^a|(x-1)$ iff $2^a|(y-1)$.
Here $a|b$ implies $a$ divides $b$.
| WLOG let $x-1=p2^a,y-1=q2^b$ where $p,q$ are odd and $a\ge b>0$
$$1+2^n=xy=(p2^a+1)(q2^b+1)=pq2^{a+b}+p2^a+q2^b+1$$
$$q=2^{n-b}-pq2^a-p2^{a-b}$$
As $n-b>0$
$q$ will be even if $a>b$
and $q$ will remain odd if $a=b$
| {
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Prove that $Z(S_n) =$ {$e$} I've read some proofs of that statement but I would like to receive a feedback on my own proof:
Let $x \in Z(S_n)$, then it particularly exchanges with a cycle of order n.
Now let $\phi$ be a cycle of order n, then:
$x\phi = \phi x$
Multiplying each side by $x^{-1}$ gives :
$x\phi x^{-1} = \... | You are missing something. Consider for example $x=\phi$ - you can't conclude that $x=id$.
What you can actually conclude is that $\phi^m$ is a cycle in $x$ for some $m\in\mathbb{Z}$.
You can use this on any two transpositions $(a,b)\ne(b,c)$ to show that $x$ fixes $a,b,c$. Alternatively you can use those transposition... | {
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Necessary and sufficient condition for the continuity of a function Let $E$ be a topological space and $A\subset E$ how to find a necessary and sufficient condition for the continuity of the function $\chi_A: E\rightarrow\mathbb{R}$ where
$$
\chi_A(x)=
\begin{cases}
1, ~x\in A\\
0,~x\not\in A
\end{cases}
$$
If i supp... | Necessary and sufficient condition for $\chi_A$ to be continuous is that $A$ is both open and closed.
Necessity. If $\chi_A$ is continuous, then $\chi_A^{-1}(-\infty,1/2)=X\setminus A$, $\chi_A^{-1}(1/2, \infty)=A$ are open.
Sufficiency. Assume that $A$ and $X\setminus A$ are both open, and $U\subset\mathbb R$ open. Th... | {
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Construct the tangent to a point on the parabola Problem:
Given the following parabola, construct a tangent to point $P$. Justify the construction.
Solution:
Draw the line $PA$ which is perpendicular to the axis and intersects that axis at $B$. Mark off the distance $OB$ and use it to find point $C$. The line $PC$ i... | EternusVia. Here is another, but NON calculus approach: Consider point of tangency $P(t,at^2)$. Tangent PC passing through P is then of the form $y=mx+at^2-mt $ (verify!).Now intersecting the tangent with the parabola gives us the equation $ax^2=mx+at^2-mt,$ which simplifies to $ax^2-mx-at^2+mt=0.$ This equation is to ... | {
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"answer_id": 3
} |
Direct Sum of Projective and Free Modules In Kevin Ventullo's answer on Mathoverflow it is mentioned that if $F$ is a free module on infinitely many generators and $P$ is a projective module, then $P \oplus F$ is again free.
However, I cannot see why this should be true.
Could you please show me why this holds?
Thank y... | I wasn't aware of this result before, and I believe as you have stated it it's false, but for some specific pairs $P$, $F$ it makes sense using the fact that a projective is a summand of a free. Specifically, let $F$ be free on $\kappa$ generators, where $\kappa$ is infinite, and suppose also that $P\oplus P' = F'$, w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2162714",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
How does the binomial expansion helps understanding graph connectedness? I read this post about graph connectedness and adjacency matrix. But I do not understand how the last comment about using the binomial expansion would help. I know this approach is correct, but just do not fully grasp it.
Would you mind to elabo... | As far as I understand $$(I+A)^{n-1} = \sum_{k=0}^{n-1} \binom{n-1}k A^k$$
Only shows how to calculate $(I+A)^{n-1}$ and there is no some extra meaning in such expansion.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2162912",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
find key problem Suppose that my keys are in the kitchen with probability $1/3$ and in the living room with probability $2/3$. If I search a room that contains the keys, then I find them with probability $0.8$. Given that I have searched the kitchen once unsuccessfully, what is the probability of the keys being in the ... | Hint: this is simply an application of Bayes' Rule .
If $K$ is "the key is in the kitchen" and $S_k$ "a search in the kitchen finds the key", then you know $\def\P{\operatorname{\mathsf P}}~\P (K)=1/3~$ and $~\P(S_k\mid K)=0.8$ (and also $~\P(S\mid K^\complement)=0~$ obviously).
Find: $~\P(K\mid S_k^\complement)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2163035",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Need hints for solving 2008 A6: $f\left(x+\frac{1}{y}\right)=f\left(y+\frac{1}{x}\right)$
Let $f(x)$ be function that satisfies
$f\left(x+\frac{1}{y}\right)=f\left(y+\frac{1}{x}\right) \Big| f:\mathbb{R} \to \mathbb{N} $.
Prove that there exists a positive integer that is not in the range of the function.
I'm aware t... | Here's a hint: you can show that in fact the function is constant on
$ \mathbb R \backslash \{ 0 \} $!
To show that, substitute $ \frac 1 y $ for $ y $ in the original equation to
get $ f ( x + y ) = f \left( \frac { x + y } { x y } \right) $
(for nonzero $ x $ and $ y $, of course). Now note that
you can choose diffe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2163304",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Closed form for the sums $S(n)={(2n-1)!\over \sqrt2}\cdot{\left(4/ \pi\right)^{2n}}\cdot\sum\limits_{k=0}^{\infty}(-1)^{k(k+1)/2}(2k+1)^{-2n}$ Consider the sums
$$S(n)={(2n-1)!\over \sqrt2}\cdot{\left(4\over \pi\right)^{2n}}\cdot\sum_{k=0}^{\infty}{(-1)^{k(k+1)\over 2}\over (2k+1)^{2n}}$$
We have $S(1)=1$, $S(2)=11$,... | We may notice that
$$ T(n)=\sum_{k\geq 0}\frac{(-1)^{k(k+1)/2}}{(2k+1)^{2n}} = \sum_{m\geq 1}\frac{\chi(m)}{m^{2n}} = L(\chi,2n)\tag{1}$$
is a Dirichlet $L$-function associated with the multiplicative function $\chi(m)$, that equals $0$ if $m$ is even, $1$ if $m\equiv \pm 1\pmod{8}$ and $-1$ if $m\equiv \pm 3\pmod{8}$.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2163401",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
Inequality for positive real numbers less than $1$: $8(abcd+1)>(a+1)(b+1)(c+1)(d+1)$ If $a,b,c,d$ are positive real numbers, each less than 1, prove that the following inequality holds:
$$8(abcd+1)>(a+1)(b+1)(c+1)(d+1).$$
I tried using $\text{AM} > \text{GM}$, but I could not prove it.
| First, note that $2^0(a+1) \ge a + 1$.
Then we prove by induction that $2^{k-1}(x_1\dots x_k+1) > \prod_{i=1}^k(x_i + 1)$. For the ease of notation, there I will focus on the case $k = 4$.
$$(d+1)(a+1)(b+1)(c+1)\le 4(d+1)(abc+1)< 4(abcd+abc+1+d)< 8(abcd+1)$$
The last inequality is provided by
$$abc+d< abcd+1 \Leftarrow... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2163467",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
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Let $a,b,c\in \Bbb R^+$ such that $(1+a+b+c)(1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c})=16$. Find $(a+b+c)$ Let $a,b,c\in \Bbb R^+$ such that $(1+a+b+c)(1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c})=16$.
Find $(a+b+c)$.
I computed the whole product ;If $(a+b+c)=x\implies (1+x)(1+\frac{bc+ca+ab}{abc})=16$. Unable to view how to p... | By $AM \ge GM$ inequality,$$(1+a+b+c)\left(1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\ge \left(4\sqrt[4]{abc}\right)\left(4\sqrt[4]{\frac{1}{abc}}\right)=16$$and equality holds when $1=a=b=c$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2163597",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Find the area of a spherical triangle made by the points $(0, 0, 1)$, $(0, 1, 0)$ and $(\frac{1}{\sqrt{2}}, 0, \frac{1}{\sqrt{2}})$.
Calculate the area of the spherical triangle defined by the points $(0, 0, 1)$, $(0, 1, 0)$ and $(\dfrac{1}{\sqrt{2}}, 0, \dfrac{1}{\sqrt{2}})$.
I have come up with this:
From the spher... | Preliminary remark: (see figure below) It is visible that the area of spherical triangle $ABC$ is a half of the area of spherical triangle $ABD$, whose area is the eighth part of the area of the sphere with radius $r=1$ ( triangle $ABD$ is the part of the sphere situated in the positive orthant). Thus, the final resul... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2163683",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
equation $x^4+ax^3-6x^2+ax+1 = 0$ has two distinct positive roots Finding parameter of $a$ for which equation $x^4+ax^3-6x^2+ax+1 = 0$
has two distinct positive roots
Attempt: writing equation $\displaystyle \bigg(x^2+\frac{1}{x^2}\bigg)+a\bigg(x+\frac{1}{x}\bigg)-6=0\;,$ where $x\neq 0$
So $\displaystyle \bigg(x+\fr... | With $\displaystyle \bigg(x+\frac{1}{x}\bigg)=t$ the roots occur when $t^2+at-8=0$.
$f(x)=x^4+ax^3-6x^2+ax+1 = 0$ shows $f(0)=1>0$ and $\displaystyle \lim_{x\to+\infty} f=+\infty$, which show $f$ has two distinct positive roots if there is $x_0>0$ for which $f(x_0)<0$. Let $\displaystyle \bigg(x_0+\frac{1}{x_0}\bigg)=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2163770",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Is there an equivalent to the Intermediate Value Theorem for $\mathbb{R}^2$ scalar fields? I am to prove some sort of mean value theorem for double integrals. That is,
if $f: R \subset \mathbb{R}^2 \rightarrow \mathbb{R}$ is continuous on some rectangle $R$, then there exists $c \in R$ such that $\iint_R f\, dA = f(c... | Yes, here it is! Mean value theorem for several variables and also for vector-valued functions.
https://en.wikipedia.org/wiki/Mean_value_theorem#Mean_value_theorem_in_several_variables
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2163922",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Number of subsets size N. So I have a problem where I have a set of ordered objects size N. For example,
A = {1, 2, 3, 4, 5}
How do I find how many ordered subsets there are of A that are size N? For instance, for N = 2, there are four total subsets of size N.
{1, 1}, {1, 2}, {2, 1}, {2, 2}
But when you make the subs... | Viewing this as a problem of stars and bars there are $N$ distinct boxes and $N$ identical balls.
$\binom{2+2-1}{2}=\binom{3}{2}=3$
$\binom{3+3-1}{3}=\binom{5}{3}=10$
$\binom{4+4-1}{4}=\binom{7}{4}=\color{red}{35}$ (you must have missed one when you were counting by hand, remember counting by hand is usually a terrible... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2164211",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Eigenvalues and corresponding eigenspace I know that $λ$ is an eigenvalue of a square matrix $A$
$\iff \exists X \ne0: AX=λX$
$\iff λ$ is a root of the characteristic polynomial of $A$.
Given a matrix $A\in \Bbb F^{n\times n}$, we find its characteristic polynomial and so its eigenvalues.
So by continuing to find corr... | As I explained before, if it is a root of the characteristic polynomial, then there exists an eigenvector; it does not matter if you can find it or not. However, if you can prove that there is no eigenvector, then it cannot be a root of characteristic polynomial.
To elevate the confusion, let me hint at a proof.
Suppos... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2164342",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
nCr and nPr: Order of Operations Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.
Where would a counting concept like nCr and nPr fall into this mix?
| There is no strong convention for these notations.
In most cases the $C$ and $P$ notations are self-delimiting, because the arguments are typeset as subscripts or superscripts. They can be written as ${}_nC_r$ or ${}^nC_r$ or $C^n_r$, where in each case it is unambiguous that the argument expression is whatever is writ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2164465",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Implicit differentiation : two or four tangents?
My Question is: The equation of a curve is $$ x ^ 2 - 2 x y + 2 y ^ 2 = 4 $$ Find the coordinates of each point on the curve at which the tangent is parallel to the $ x $-axis.
So I established that $$ \frac { \mathop { \sf d } y } { \mathop { \sf d } x } = 0 \implies ... | Subbing $x$ into the original equation isn't wrong, it's just not enough. This is because $x^2 - 2xy + 2y^2 = 4$ does not express $y$ as a function of $x$.
With implicit functions, it's not enough to just plug the $x$-coordinates back into the original function, because that doesn't tell you which $x$- and $y$-coordin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2164599",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Finding The Derivative Using $\frac{d}{dx}x^n=nx^{n-1}$ So I am learning how to differentiate now,
and I came across this problem
$$f(x)=\frac{1-x}{2+x}$$
We are wanted to find $f'(x)$.
When I use $$\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$$
I find that $f'(x)=\frac{-3}{(2+x)^2}$ but, when I try to find $f'(x)$ the easy way.... | You can try this
$\frac{d}{dx}f(x) = \frac{d}{dx}(-1+\frac{3}{2+x}) = 3\frac{d}{dx}(2+x)^{-1} = \frac{-3}{(x+2)^2}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2164704",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
Diffeq question: method of undetermined coefficients I have a question here from Differential Equations. We are learning how to solve second order nonhomogenous equations using the method of undetermined coefficients. The equation at hand is $$y''+4y=3\sin(2t)$$
I understand that the solution to the corresponding homo... | Your problem is $(D^2+4)[y] = 3 \sin (2t)$. Operate by $D^2+4$ once again to find $(D^2+4)^2[y] = (D^2+4)(3\sin(2t)) =0$. Then,
$$ y = c_1 \cos( 2t)+c_2\sin( 2t) + c_3t\cos (2t) + c_4t \sin (2t) $$
is the general solution. Identify the first two terms as the homogeneous solution to the given differential equation. Henc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2164800",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Solving $\int_0^1 \frac{e^{-t} }{1+t} dt\,$ using Exponential Integral I'm having a really hard time figuring out how to solve the following integrals:
$$
\int_0^1 \frac{e^{-t} }{1+t} dt\, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\int_0^1 \frac{e^{-t} }{(1+t)^2} dt\,
$$
According to my professor, the m-th Exponential Integral ... | How about using the geometric series? That is
$$\frac{1}{1+t}=\sum_{n=0}^\infty (-t)^n$$
Then we have
$$
\int^1_0\frac{e^{-t}}{1+t}\,dt = \sum_{n=0}^\infty \int^1_0 e^{-t}(-t)^n\,dt.
$$
This isn't quite the exponential integral, since the power of $t$ is in the numerator instead of the denominator, and the bounds are ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2164882",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Calculate the following limit I need to evaluate the following limit, however, in doing so, I let $\sqrt{1+c}=1$ which I came to undertand that it's not valid. My procedure was the following:
$$\lim_{c\to0}\left(-\ln(c)\sqrt{1+c}-\ln\left(\frac{1+\sqrt{1+c}}{1-\sqrt{1+c}}\right)+\ln\left(\frac{1+\sqrt2}{1-\sqrt2}\right... | You can write this as
$$-\ln(c)\sqrt{1+c}-\ln\left(\frac{1+\sqrt{1+c}}{1-\sqrt{1+c}}\right)+\ln\left(\frac{1+\sqrt2}{1-\sqrt2}\right) \\
= -\ln(c)\sqrt{1+c} + \ln c- \ln c -\ln\left(\frac{1+\sqrt{1+c}}{1-\sqrt{1+c}}\right)+\ln\left(\frac{1+\sqrt2}{1-\sqrt2}\right) \\ = \ln c ( 1 - \sqrt{1 + c}) + A(c) $$
where
$$A(c)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2164990",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Is any multiplicative linear functional $ \Phi $ on $ C([a,b]) $ bounded? Denote by $ X $ the function space
$$
C([a,b])
= \{
u: [a,b] \to \mathbb{C} \mid u ~ \text{is continuous at each} ~ t \in [a,b]
\}
$$
over the scalar field $ \mathbb{C} $ of complex numbers, as usual, in which the norm of $ u \in X $ is giv... | Here is a more direct proof of your original question without proving your guess, or using measure theory.
*
*$\Phi(1)=1$, where $1$ denotes the constant function with value $1$.
*For any function $x\in C[a,b]$ without zeroes, it is $\Phi(x)\neq 0$, because $\Phi(x)\Phi(x^{-1})=\Phi(xx^{-1})=\Phi(1)=1$, where $x^{-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2165065",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Prove a relation between functions from an inequality relating limits Given two functions $f$ and $g$ satisfying the inequality $$\lim_{h \to 0} \frac{f(h)}{h} > \lim_{h \to 0} \frac{g(h)}{h} \tag{1},$$
prove that there exists $h>0$ such that $f(h) > g(h)$. I was thinking that intuitively, if $f(0) = g(0)$ and $f'(0) >... | Intuitively, you have two intervals,
$$\left(A - \varepsilon, A + \varepsilon\right) \text{ and } \left(B - \varepsilon,B + \varepsilon\right),$$
and if you take $|h|$ sufficiently small (in your case, if you take $|h|<\delta_1$ and $|h|<\delta_2$),
$$\frac{f(h)}{h} \in \left(A - \varepsilon, A + \varepsilon\right) \te... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2165175",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Series rapresentation $f(t)=\frac{1}{(2t+1)^{2}}$ How to find the series rapresentation in $t=0$ of the following expression
$$f(t)=\frac{1}{(2t+1)^{2}}$$
Can someone show me also the steps.
Thank you so much!
| $\displaystyle \frac{1}{1-x}=\sum\limits_{n=0}^\infty x^n\enspace$ for $\enspace-1<x<1$ , proof by multiplication with $1-x$ .
One derivation for $x$ gives $\enspace\displaystyle \frac{1}{(1-x)^2}=\sum\limits_{n=1}^\infty nx^{n-1}$ .
With $x:=-2t$ and therefore $\enspace -\frac{1}{2}< t< \frac{1}{2}$ follows:
$$\disp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2165286",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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