qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
3,363,944 | <p>A group consisting of <span class="math-container">$3$</span> men and <span class="math-container">$6$</span> women attends a prizegiving ceremony. If <span class="math-container">$ 5$</span> prizes are awarded at random to members of the group, find the probability that exactly <span class="math-container">$3 $</sp... | Avinash N | 253,506 | <p>Let's start.( Answer for b)</p>
<p>Total number of prizes <span class="math-container">$=5$</span>.</p>
<p>The number of ways to select <span class="math-container">$3$</span> prizes from <span class="math-container">$5$</span> prizes <span class="math-container">$=5C3$</span> <span class="math-container">$=10$</s... |
2,302,067 | <p>I'm trying to prove that if ${\kappa}$ is an infinite cardinal, then there are $2^{\kappa}$ bijective functions from ${\kappa}$ to ${\kappa}$. I would greatly appreciate any tips. Thank you. </p>
| bof | 111,012 | <p>We know that there are at most $\kappa^\kappa\le(2^\kappa)^\kappa=2^{\kappa^2}=2^\kappa$ bijections from $\kappa$ to $\kappa;$ we have to show that there are at least $2^\kappa$ bijections. Since $|\{0,1\}\times\kappa|=2\kappa=\kappa,$ it will suffice to exhibit $2^\kappa$ bijections from $\{0,1\}\times\kappa$ to $\... |
3,702,086 | <h1>Problem</h1>
<p>In a two-dimensional Cartesian coordinate system,
there are two points <span class="math-container">$A(2, 0)$</span> and <span class="math-container">$B(2, 2)$</span>
and a circle <span class="math-container">$c$</span> with radius <span class="math-container">$1$</span> centered at the origin <spa... | user | 293,846 | <p>Your hypothesis is true.</p>
<p>Indeed the solution of the equation:
<span class="math-container">$$
2\sqrt{10-8x}+\sqrt{9-4x-4\sqrt{1-x^2}}=5
$$</span>
is
<span class="math-container">$$
x=\frac{2+7\sqrt{46}}{50},\text{ with } \sqrt{1-x^2}=\frac{14-\sqrt{46}}{50}.
$$</span></p>
<p>Substituting this into the deriv... |
3,702,086 | <h1>Problem</h1>
<p>In a two-dimensional Cartesian coordinate system,
there are two points <span class="math-container">$A(2, 0)$</span> and <span class="math-container">$B(2, 2)$</span>
and a circle <span class="math-container">$c$</span> with radius <span class="math-container">$1$</span> centered at the origin <spa... | Claude Leibovici | 82,404 | <p>The problem is interesting for sure but I believe that the solution is biased by the fact that <span class="math-container">$x_A=x_B=y_A$</span>.</p>
<p>I tried to make it more general with <span class="math-container">$A(x_A,y_A)$</span> and <span class="math-container">$B(x_B,0)$</span> assuming that both points ... |
4,317,945 | <p>A function <span class="math-container">$h : A → \mathbb{R}$</span> is Lipschitz continuous if <span class="math-container">$\exists K$</span> s.t.</p>
<p><span class="math-container">$$|h(x) - h(y)| \leq K \cdot |x - y|, \forall x, y \in A$$</span></p>
<p>Suppose that <span class="math-container">$I = [a, b]$</span... | paul garrett | 12,291 | <p>A small amount of cleverness makes this much more tractable, and avoids combinatorics:</p>
<p>Namely, the leading factor of <span class="math-container">$x^2$</span> is <span class="math-container">$(x+1)^2-2(x+1)+1$</span>. So the whole expression can be expressed in powers of <span class="math-container">$x+1$</sp... |
215,835 | <p>According to Willard,</p>
<p>If $(X,\tau)$ is a topological space, a base for $\tau$ is a collection $\mathscr{B} \subset \tau$ such that $\tau=\{ \bigcup_{B \in \mathscr C} : \mathscr C \subset \mathscr B\}$. Evidently, $\mathscr B$ is a base for $X$ iff whenever $G$ is an open set in $X$ and $p \in G$ there is so... | Berci | 41,488 | <p>Q1: Base sets are <strong>open</strong>, indeed: consider the one element $\mathscr C$ subsets.</p>
<p>Q2: Lied. Or thought about something else..</p>
<p>A system $\mathscr B$ of subsets of a set $X$ can be a <em>basis</em> for a topology iff $\forall B,C\in\mathscr B\ \forall x\in B\cap C\ \exists D\in\mathscr B$... |
215,835 | <p>According to Willard,</p>
<p>If $(X,\tau)$ is a topological space, a base for $\tau$ is a collection $\mathscr{B} \subset \tau$ such that $\tau=\{ \bigcup_{B \in \mathscr C} : \mathscr C \subset \mathscr B\}$. Evidently, $\mathscr B$ is a base for $X$ iff whenever $G$ is an open set in $X$ and $p \in G$ there is so... | Brian M. Scott | 12,042 | <p><strong>Question 1:</strong> Yes, Willard was still talking about collections $\mathscr{B}\subseteq\tau$. However, with a minor change in wording he wouldn’t have to be, because we could deduce that $\mathscr{B}\subseteq\tau$. Specifically, suppose that $\langle X,\tau\rangle$ is a topological space, and that $\math... |
1,859,719 | <blockquote>
<p>Let be $U (x,y) = x^\alpha y^\beta$. Find the maximum of the function $U(x,y)$ subject to the equality constraint $I = px + qy$.</p>
</blockquote>
<p>I have tried to use the Lagrangian function to find the solution for the problem, with the equation</p>
<p>$$\nabla\mathscr{L}=\vec{0}$$</p>
<p>where... | marty cohen | 13,079 | <p>If
$I=px+qy$,
then
$y = (I-px)/q$,
so
$x^ay^b
=x^a((I-px)/q)^b
=x^a(I-px)^b/q^b
$.</p>
<p>Differentiating,
we want</p>
<p>$\begin{array}\\
0
&=(x^a(I-px)^b)'\\
&=ax^{a-1}(I-px)^b-x^apb(I-px)^{b-1}\\
&=x^{a-1}(I-px)^{b-1}(a(I-px)-xpb)\\
&=x^{a-1}(I-px)^{b-1}(aI-apx-xpb)\\
&=x^{a-1}(I-px)^{b-1}(a... |
1,878,884 | <p>I recently figured out my own algorithm to factorize a number given we know it has $2$ distinct prime factors. Let:</p>
<p>$$ ab = c$$</p>
<p>Where, $a<b$</p>
<p>Then it isn't difficult to show that:</p>
<p>$$ \frac{c!}{c^a}= \text{integer}$$</p>
<p>In fact, </p>
<p>$$ \frac{c!}{c^{a+1}} \neq \text{integer}... | gt6989b | 16,192 | <p>Not sure about correctness, but since $c$ has a representation in $\log c$ bits, you have to make $\Theta(c)$ multiplications to do this naively, so this algorithm is <strong>expoential</strong>, not polynomial</p>
<p><strong>UPDATE</strong></p>
<p>The edit improves on the number of divisions, but not on the numbe... |
2,021,557 | <p>I'm not really sure how to do this, I guessed it had something to do with Vector Functions but overall couldn't find a way to do it. Can you please help?</p>
<p>The equations are:</p>
<p>$$f(x,y) = x^2 + y^2
\ g(x,y) = xy + 10 $$</p>
<p>and I need a Vectorial equation.
Thank you in advance!</p>
| Jean Marie | 305,862 | <p>We look for the intersection of the two surfaces (which are a paraboloid and a hyperbolic paraboloid):</p>
<p>$$\tag{0}\cases{z=x^2+y^2 & (a)\\z=xy+10 & (b)}$$</p>
<p>Let $(r,\theta)$ be the polar coordinates of $(x,y)$; i.e, </p>
<p>$$\tag{1}x=r \cos(\theta), \ \ y=r \sin(\theta).$$</p>
<p>Plugging th... |
1,102,885 | <p>I have exams in Machine Learning coming up and I need help answering this question.</p>
<blockquote>
<p>There are a million identical fish in a lake, one of which has
swallowed the One True Ring. You must get it back! After months of
effort, you catch another random fish and pass your metal detector
over it... | Aerinmund Fagelson | 173,945 | <p>Surely the probability of the detector beeping if you have found the fish is 999999999/1000000000 not 9999/10000
Whereas the probability of the detector not beeping when the fish is not found is 9999/10000 not 999999999/1000000000?</p>
|
3,914,626 | <p>Let <span class="math-container">$A$</span> be a <span class="math-container">$k$</span>-dimensional non singualar matrix with integer coefficients. Is it true that <span class="math-container">$\|A^{-1}\|_\infty \leq 1$</span>? How can I show that? Could you give me a counterexample?It is clear that <span class="ma... | Bram28 | 256,001 | <p>First of all, I hope you understand the intuition behind this:</p>
<p>Just because some <em>specific</em> object has some property obviously does not mean that <em>all</em> objects from the domain have that property.</p>
<p>However, if an arbitrary object from the domain has some property, then all objects do.</p>
<... |
1,347 | <p>Sometimes I check how many users of <code>mathematica.stackexchange.com</code> there are.<br>
I remember that a few weeks ago there were about 15 thousand and recently I've been surprised seeing that the <a href="https://mathematica.stackexchange.com/users?tab=NewUsers&sort=creationdate">new users</a> are signe... | Oded | 6,870 | <p>This is due to how our databases are setup and operate.</p>
<p>The Id field (user number) is an auto incrementing field - essentially when a user gets created, it takes the next number (not entirely accurate, there are some wrinkles there, not relevant to this).</p>
<p>We operate with replication - our databases a... |
1,457,956 | <p>I am finding the positive values of $x$ for which the following series is convergent $$ \sum_{n=1}^{\infty}x^{\sqrt{n}}$$ It is sure that it is not convergent for $x\geq1$ as $n$-th term will not tend to zero. Now $x\in[0,1)$ how to check its convergence? Please help me to solve it. Thanks. </p>
| Yes | 155,328 | <p>Let $x \in ]0,1[$; then
$$x^{\sqrt{n}} = \exp(\sqrt{n} \log x) = \frac{1}{e^{\sqrt{n}|\log x|}} < \frac{1}{(\sqrt{n})^{3}}
$$
for large $n$, so by the comparison test the desired series converges.</p>
|
1,457,956 | <p>I am finding the positive values of $x$ for which the following series is convergent $$ \sum_{n=1}^{\infty}x^{\sqrt{n}}$$ It is sure that it is not convergent for $x\geq1$ as $n$-th term will not tend to zero. Now $x\in[0,1)$ how to check its convergence? Please help me to solve it. Thanks. </p>
| Bernard | 202,857 | <p><em>Without the exponential:</em></p>
<p>If $0\le x <1$, we have:
\begin{align*}
\sum_{n\ge1}x^{\sqrt n}&\le\sum_{n\ge1}x^{\lfloor\sqrt n\rfloor}=3x+5x^2+7x^3+9x^4+\dotsm \\
&\le 2+4x+6x^2+8x^3+10x^4+\dotsm=2(1+2x+3x^2+4x^3+5x^4+\dotsm)\\
&= 2\biggl(\frac1{1-x}\biggr)'=\frac2{(1-x)^2}
\end{align*}
he... |
802,960 | <p>$$\sum\limits_{k=1}^n\arctan\frac{ 1 }{ k }=\frac{\pi}{ 2 }$$
Find value of $n$ for which equation is satisfied. </p>
| Tom-Tom | 116,182 | <p>Let use write $$s_n=\sum_{k=1}^n \arctan\frac1k.$$
The sequence $(s_n)_{n\in\mathbf N}$ is increasing.
We have $s_0=0$, $s_1=\frac\pi4$ and $s_2=\frac\pi4+\arctan\frac12$.
As $\frac12<1$, $\tan^{-1}\left(\frac12\right)<\frac\pi4$ and $s_2<\frac\pi2$.
Let us compute $s_3$ using the <a href="http://en.wikiped... |
874,300 | <p>I'm having trouble grasping how to set these types of problems. There are a lot of related questions but it's difficult to abstract a general procedure on finding constants that give the given function bounding constraints to make it big-theta(general function). </p>
<p>so $\frac{x^4 +7x^3+5}{4x+1}$ is $ \Theta ... | brogrenkp | 115,493 | <p>You are on the right track. However, rather than dividing by $x^3$, I would recommend multiplying by $(4x+1)$. The reason for this is so that you will have polynomials of degree $4$ on all sides of the inequality.</p>
<p>It is okay to try different values for $k$ once you get a more simplified inequality. For this ... |
373,958 | <p>Is $\sum_{n=1}^\infty(2^{\frac1{n}}-1)$ convergent or divergent?
$$\lim_{n\to\infty}(2^{\frac1{n}}-1) = 0$$
I can't think of anything to compare it against. The integral looks too hard:
$$\int_1^\infty(2^{\frac1{n}}-1)dn = ?$$
Root test seems useless as $\left(2^{\frac1{n}}\right)^{\frac1{n}}$ is probably even harde... | Paolo Leonetti | 45,736 | <p>Since $2^x=1+x\ln 2+O(x^2)$ as $x\to 0$ then
$$\sum_{n\ge 1}\left(2^{1/n}-1\right)\asymp \sum_{n\ge 1}\frac{1}{n},$$
which diverges.</p>
|
373,958 | <p>Is $\sum_{n=1}^\infty(2^{\frac1{n}}-1)$ convergent or divergent?
$$\lim_{n\to\infty}(2^{\frac1{n}}-1) = 0$$
I can't think of anything to compare it against. The integral looks too hard:
$$\int_1^\infty(2^{\frac1{n}}-1)dn = ?$$
Root test seems useless as $\left(2^{\frac1{n}}\right)^{\frac1{n}}$ is probably even harde... | paw88789 | 147,810 | <p>You could use the fact that for a series of positive terms, $\sum_{n=1}^\infty a_n$ converges if and only if $\prod_{n=1}^\infty (1+a_n)$ converges.</p>
<p>Applying this result to the given problem: The given series converges if and only if the infinite product $\prod_{n=1}^\infty 2^{\frac1n}$</p>
<p>For this infi... |
91,590 | <p>So I'm reviewing old homeworks for an upcoming comp sci test and I came across this question:</p>
<p>Say whether the following statement is True, False or Unknown: </p>
<blockquote>
<p>The problem of checking whether a given Boolean formula has exactly
one satisfying assignment, is NP-complete</p>
</blockquote... | templatetypedef | 8,955 | <p>I believe that this is an open problem because I think that the problem of "does φ have exactly one satisfying assignment?" is, I believe, co-NP-complete by a reduction from the unsatisfiability problem, which is known to be co-NP-complete. The idea is that given a formula φ with variables v<sub>1</sub>, v<... |
1,672,131 | <p>A card game is played with a deck whose cards can be one of 6 suits, one of the suits being hearts, and one of 11 ranks. A hand is a subset of 3 cards. What is the probability that a hand has exactly two hearts given that it has the 2 of hearts? Please explain.</p>
| Robert Israel | 8,508 | <p>Hint: $x$ has $n$ digits if $10^{n-1} \le x < 10^n$.</p>
|
2,282,818 | <p>I'm getting $f(x)=2x+f(0)$ and $f(x)=f(0)-2x$ by setting $y=0$, but I'd like to verify. Am I right?</p>
| Martin R | 42,969 | <p>For <span class="math-container">$y = 0$</span> we get that
<span class="math-container">$$
f(x) = f(0) \pm 2x
$$</span>
for all <span class="math-container">$x \in \Bbb R$</span>. We want to show that the same sign must hold for all <span class="math-container">$x$</span>, i.e. either
<span class="math-container">... |
2,619,185 | <p>Let $$P=(X+2)^m+(X+3)^{2m+3}$$ and $$Q=X^2+5X+7.$$ I need to show that $Q$ divides $P$ for any $m$ natural. </p>
<p>I said like this: let $a$ be a root of $X^2+5X+7=0$. Then $a^2+5a+7=0$. </p>
<p>Now, I know I need to show that $P(a)=0$, but I do not know if it is the right path since I have not found any way to d... | zwim | 399,263 | <p>We can also prove it by induction.</p>
<p>$P_0(x)=1+(x+3)^3=x^3+9x^2+27x+28=(x^2+5x+7)(x+4)\quad\checkmark$</p>
<p>$\begin{align}
P_{m+1}(x) &=(x+2)^{m+1}+(x+3)^{2m+5}\\
&=(x+2)^m(x+2)+(x+3)^2\overbrace{\big((x^2+5x+7)Q_m(x)-(x+2)^m\big)}^{\text{induction hypothesis}}\\\\
&=(x+2)^m\underbrace{(x+2-x^2-... |
804,882 | <p>If both $L:V\rightarrow W$ and $M:W\rightarrow U$ are linear transformations that are invertible, how can you prove that the composition $(M\circ L):V\rightarrow U$ is also invertible.</p>
| EPS | 133,563 | <p>Composition of two invertible functions is invertible and composition of two linear maps is linear.</p>
|
181,367 | <p>It is well known that compactness implies pseudocompactness; this follows from <a href="https://secure.wikimedia.org/wikipedia/en/wiki/Heine%E2%80%93Borel_theorem">the Heine–Borel theorem</a>. I know that the converse does not hold, but what is a counterexample?</p>
<p>(A <a href="https://secure.wikimedia.org/wikip... | MJD | 25,554 | <p>I couldn't think of an obvious counterexample, so I looked <a href="https://secure.wikimedia.org/wikipedia/en/wiki/Pseudocompact_space" rel="noreferrer">in Wikipedia</a> and it suggested <a href="https://secure.wikimedia.org/wikipedia/en/wiki/Particular_point_topology" rel="noreferrer">the particular point topology<... |
181,367 | <p>It is well known that compactness implies pseudocompactness; this follows from <a href="https://secure.wikimedia.org/wikipedia/en/wiki/Heine%E2%80%93Borel_theorem">the Heine–Borel theorem</a>. I know that the converse does not hold, but what is a counterexample?</p>
<p>(A <a href="https://secure.wikimedia.org/wikip... | user3810316 | 426,834 | <p>This nice paper presents all kinds of relations between different compactness notions.
<a href="http://www.cs.cmu.edu/~yaoliang/mynotes/compact.pdf" rel="nofollow noreferrer">http://www.cs.cmu.edu/~yaoliang/mynotes/compact.pdf</a></p>
<p>E.g., if a topological space $K$ is compact or sequentially compact, then it i... |
160,518 | <p>In Mathematics, we know the following is true:</p>
<p>$$\int \frac{1}{x} \space dx = \ln(x)$$</p>
<p>Not only that, this rule works for constants added to x:
$$\int \frac{1}{x + 1}\space dx = \ln(x + 1) + C{3}$$
$$\int \frac{1}{x + 3}\space dx = \ln(x + 3) + C$$
$$\int \frac{1}{x - 37}\space dx = \ln(x - 37) + C$$... | Community | -1 | <p>Perhaps I can reverse-address your question. Oftentimes (typically in optimization problems) when dealing with a positive real function $f$ it is easier to differentiate $\log f$ than $f$ itself. It's easy to check that the so-called logarithmic derivative satisfies $\frac{d}{dx} \log [f (x)] = \frac{f'(x)}{f(x)}.$ ... |
3,529,359 | <p>Let <span class="math-container">$\Omega$</span> be a bounded and smooth domain and let <span class="math-container">$J:H^1(\Omega) \times H^1_0(\Omega) \to \mathbb{R}$</span> be defined by</p>
<p><span class="math-container">$$J(u,v) = \int_\Omega f(u)|\nabla v|^2$$</span>
where <span class="math-container">$f\col... | Johannes Hahn | 62,443 | <p>First note that <span class="math-container">$f(u)\|\nabla v\|^2 = \|\sqrt{f(u)}v\|^2$</span>. Therefore your integral is <span class="math-container">$J(u,v)=\|\sqrt{f(u)}v\|_{L^2}^2$</span>. Now it is known that <span class="math-container">$x_n \xrightarrow[weak]{X} x \implies \liminf \|x_n\|_X \geq \|x\|_X$</spa... |
2,603,239 | <p>(The Cauchy principal value of)
$$
\int_0^{\infty}\frac{\tan x}{x}\mathrm dx
$$</p>
<p>I tried to cut this integral into $$\sum_{k=0}^{\infty}\int_{k\pi}^{(k+1)\pi}\frac{\tan x}{x}\mathrm dx$$
And then
$$\sum_{k=0}^{\infty}\lim_{\epsilon \to 0}\int_{k\pi}^{(k+1/2)\pi-\epsilon}\frac{\tan x}{x}\mathrm dx+\int_{(k+1/2... | spaceisdarkgreen | 397,125 | <p>You seem to be on the right track.</p>
<p>We have $$ P\int_0^\infty \frac{\tan{x}}{x}dx = P\int_0^\pi \frac{\tan x}{x}dx + P\int_0^\pi\frac{\tan x}{\pi + x}dx + P\int_0^\pi \frac{\tan x}{2\pi + x}dx+\ldots$$ and then we have $$ P\int_0^\pi \frac{\tan x}{k\pi +x}dx = \int_0^{\pi/2} \tan x\left(\frac{1}{k\pi+x} - \fr... |
3,414,197 | <p>I have to model/simulate a moving iron meter with Simulink, more specifically, I need to build a Simulink model for the equation of motion, wich is given as:
<span class="math-container">$$
\theta\ddot{\alpha} = T_\phi - T_S
$$</span>
where <span class="math-container">$\theta$</span> denotes the pointers moment ... | Dinno Koluh | 519,191 | <p>I would personally do the problem in an other way. I guess that your input is the DC voltage and the output is the angle. The differential equation of the system as you wrote it is:
<span class="math-container">$$
\theta \ddot{\alpha}(t) = \frac{c_\phi}{R}v(t) -\frac {c_\phi c_i}{R}\dot{\alpha}(t)-c_S\alpha(t)
$$</s... |
3,414,197 | <p>I have to model/simulate a moving iron meter with Simulink, more specifically, I need to build a Simulink model for the equation of motion, wich is given as:
<span class="math-container">$$
\theta\ddot{\alpha} = T_\phi - T_S
$$</span>
where <span class="math-container">$\theta$</span> denotes the pointers moment ... | Pilotf4 | 272,311 | <p><strong>EDIT:</strong> Had some typos in my transfer function and now I get the same results from my transfer function model! </p>
<p>Thank you for your answer!
I tried to compare the two forms, but I think something must've gone wrong.
For , I get <span class="math-container">$\frac{c_\phi c_i}{2R \theta \omega_0... |
2,060,156 | <p>First thing I want to mention is that this is not a topic about why $1+2+3+... = -1/12$ but rather the connection between this summation and $\zeta$.</p>
<p>I perfectly understand that the definition using the summation $\sum_{k=1}^\infty k^{-s}$ of the zeta function is only valid for $Re(s) > 1$ and that the fu... | Vidyanshu Mishra | 363,566 | <p>Suppose there is integer $p$ which can be written as $\frac{6l-1}{4l-3}$ and $\frac{7k-5}{5k-3}$. </p>
<p>$$p= \frac{6l-1}{4l-3} =\frac{7k-5}{5k-3}$$</p>
<p>$$\implies kl+8k+l=6$$</p>
<p>$$\implies(k+1)l=(6-8k)\implies l=\frac{-2(4k-3)}{(k+1)}$$.</p>
<p>Which gives following integer solutions:</p>
<p>$(k,l)=(-1... |
1,151,653 | <p>How can I express the following as a function sequence? Namely, how can I properly express <span class="math-container">$f_n(x)$</span>?</p>
<p>Here are the following function graphs:</p>
<p><img src="https://i.stack.imgur.com/2GFYj.png" alt="enter image description here" /></p>
<p>Text only (color-coded with image)... | Community | -1 | <p>Hint: $$x^2+y^2=1$$
$$y=x^2$$</p>
<p>Where do they intersect?</p>
|
1,384,735 | <p>What is the ODE satisfied by $y=y(x)$ </p>
<p>given that $$\frac{dy}{dx} = \frac{-x-2y}{y-2x}$$</p>
<p>I understand that I need to get it in some form of $\int \cdots \;dy = \int \cdots \; dx$, but am not sure how to go about it.</p>
| Dr. Sonnhard Graubner | 175,066 | <p>rewrite your equation in the form $$\frac{dy}{dx}=\frac{-1-2\frac{y}{x}}{\frac{y}{x}-2}$$ and set $$y=xu$$</p>
|
3,973,006 | <p>The question is fully contained in the title.</p>
<p>I tried to prove maximality (if that happens, <span class="math-container">$I$</span> is prime as well) in <span class="math-container">$\mathbb Z[X]$</span>, but I am not able to figure a strategy out for that purpouse. Obviously, if <span class="math-container">... | Arthur | 15,500 | <p>Let's just play around and see what kinds of polynomials we can find in <span class="math-container">$I$</span>.</p>
<p>First of all, we can try to cancel the cubic term from the cubic generator, and we see that
<span class="math-container">$$
7(X^3+2X^2+1)-X^2(7X+14)=7
$$</span>
is an element of the ideal. And sinc... |
316,865 | <p>How do you find this limit?</p>
<p>$$\lim_{x \rightarrow \infty} \sqrt[5]{x^5-x^4} -x$$</p>
<p>I was given a clue to use L'Hospital's rule.</p>
<p>I did it this way:</p>
<p><strong>UPDATE 1:</strong>
$$
\begin{align*}
\lim_{x \rightarrow \infty} \sqrt[5]{x^5-x^4} -x
&= \lim_{x \rightarrow \infty} x\begin{p... | Mikasa | 8,581 | <p>You got the answer, but I'd like to note something different. I see you are doing derivations, so I am writing an answer based on it. We say the function $\alpha(x)$ is very small at $x\to a$ when $$\lim\alpha(x)\to 0$$ We can prove that by using Taylor expansion that $\sqrt[n]{1+\alpha(x)}-1\sim\frac{\alpha(x)}{n}$... |
1,695,261 | <p>Is it true that for every $ε > 0$, there is $δ > 0$, such that $0 < |x−2| < δ ⇒ |(x^2 −x)−2| < ε$?</p>
<p>Now I know that $|(x^2 −x)−2|$ is same as $|(x-2)(x+1)|$, but I am not sure how to link that with the first bit of info given. In general epsilon-delta proofs confuse me. </p>
<p>So I start by s... | crbah | 314,622 | <p>Let the $\epsilon = \epsilon_0$ satisfying $|x^2-x-2| < \epsilon_0$. Initially choose $\delta$ to be $1$. We will refine this delta.</p>
<p>$-\epsilon_0 < x^2-x-2 < \epsilon_0$</p>
<p>$\implies -\epsilon_0+\frac{9}{4} < x^2-x+\frac{1}{4} < \epsilon_0+\frac{9}{4}$</p>
<p>$\implies -\epsilon_0+\frac{... |
1,695,261 | <p>Is it true that for every $ε > 0$, there is $δ > 0$, such that $0 < |x−2| < δ ⇒ |(x^2 −x)−2| < ε$?</p>
<p>Now I know that $|(x^2 −x)−2|$ is same as $|(x-2)(x+1)|$, but I am not sure how to link that with the first bit of info given. In general epsilon-delta proofs confuse me. </p>
<p>So I start by s... | DanielWainfleet | 254,665 | <p>Here are some general results which enable us to handle the Q of continuity for a broad class of real functions: </p>
<p>(1)... Constant functions are continuous.</p>
<p>(2)... f(x)=x is continuous </p>
<p>(3)...f(x)=|x| is continuous.</p>
<p>(4)...For continuous f, g :</p>
<p>... (i)... h(x)=f(x)+g(x) is cont... |
1,376,159 | <p>A friend of mine shared this problem with me. As he was told, this integral can be evaluated in a closed form (the result may involve polylogarithms). Despite all our efforts, so far we have not achieved anything, so I decided to ask for your advice.
$$\int_0^1\log(x)\,\log(2+x)\,\log(1+x)\,\log\left(1+x^{-1}\right)... | Start wearing purple | 73,025 | <p>The main ingredient here is the integral representation
$$\operatorname{Li}_n(z)=\frac{(-1)^{n-1}}{(n-2)!}\int_0^1
\frac{\ln\left(1-zx\right)\ln^{n-2}x\,dx}{x},\tag{$\spadesuit$}$$
valid for $|z|<1,n\in\mathbb{N}_{\ge 2}$.</p>
<p>The derivation goes as follows:</p>
<ol>
<li><p>Rewrite the initial integral as
\... |
2,037,704 | <p>What symmetry property in complex space is related to the fact that the absolute value of numbers $|a+ib| = |b+ia|$ are equals?</p>
| GEdgar | 442 | <p>In $\mathbb R^2$, the map $(a,b) \mapsto (b,a)$ is reflection in the
45-degree line $y=x$. This map is (of course) an isometry of the plane, so it is an isometry of $\mathbb C$.</p>
|
3,470,208 | <p><span class="math-container">$$f(x)=\begin{cases}
\dfrac{x}{\sin x}, & x>0\\
2-x, & x\le0
\end{cases}$$</span></p>
<p><span class="math-container">$$g(x)=\begin{cases}
x+3, &x<1\\
x^2-2x-2, &1\le x<2\\
x-5, & x\ge2
\end{cases}$$</span></p>
<p>Find left hand limi... | user3290550 | 278,972 | <p>I didn't compose the functions in the proper way, thanks to @user for pointing it out.</p>
<p>I found an interesting way to compose it which will avoid mistakes.</p>
<p>Replace <span class="math-container">$x$</span> by <span class="math-container">$f(x)$</span></p>
<p><span class="math-container">$$g(f(x))=\begi... |
2,755,143 | <p>Find Number of integers satisfying $$\left[\frac{x}{100}\left[\frac{x}{100}\right]\right]=5$$ where $[.]$ is Floor function.</p>
<p>I assumed $$x=100q+r$$ where $0 \le r \le 99$</p>
<p>Then we have </p>
<p>$$\left[\left(q+\frac{r}{100}\right)q\right]=5$$ $\implies$</p>
<p>$$q^2+\left[\frac{rq}{100}\right]=5$$</... | Frostic | 402,923 | <p>I got $x\in [|250,299|]$ </p>
<p>I solved it writing $x = a10^2+b10^1+c10^0$. And reasoning on $a$ then $b$ then $c$ given the fact that $f$ is non decreasing. </p>
<p>$f(200) = 4$ and
$f(300) = 9$</p>
<p>Therefore $a = 2$</p>
<hr>
<p>$f(240) = 4$ and
$f(250) = 5$ and
$f(290) = 5$</p>
<p>Therefore $5\leq b \l... |
3,519,515 | <p>Here, I wonder what is a good way to use the epsilon delta definition or converging sequences to show that the set S containing quotients on [0,1] have/does not have volume 0, (i.e. whether there exist a <strong>finite</strong> number of intervals which union contain all of S such that the <strong>sum</strong> of le... | Sarvesh Ravichandran Iyer | 316,409 | <p>Indeed, closure does play a part, if you are going for <em>finitely</em> many intervals to do the covering.</p>
<p>The point is, if <span class="math-container">$I_1,...,I_n$</span> are intervals that covered <span class="math-container">$S$</span>, say <span class="math-container">$S \subset \cup_{i=1}^n I_i$</spa... |
6,931 | <p>One of the key steps in <a href="http://en.wikipedia.org/wiki/Merge_sort">merge sort</a> is the merging step. Given two sorted lists</p>
<pre><code>sorted1={2,6,10,13,16,17,19};
sorted2={1,3,4,5,7,8,9,11,12,14,15,18,20};
</code></pre>
<p>of integers, we want to produce a new list as follows:</p>
<ol>
<li>Start w... | Leonid Shifrin | 81 | <h2>Preamble</h2>
<p>Since I agree that it would be nice to have a generic function of this type, I will provide a general implementation. First, I will give a generic one based on linked lists, then I will add a JIT-compiled one for special numeric types, and lastly, I will bring it all together.</p>
<h2>Top-level i... |
58,525 | <p>I am trying to make surface plots of squashed spheres. The spheres are defined by a list of points. For simplicity, consider the round sphere:</p>
<pre><code>pts = Flatten[
Table[{Sin[θ] Cos[ϕ], Sin[θ] Sin[ϕ],
Cos[θ]}, {θ, 0, π, π/14}, {ϕ, 0,
2 π, 2 π/14}], 1];
</code></pre>
<p>One way to plot thi... | Jens | 245 | <p>From the example using <code>ListPlot3D</code>, I assume that your data points can be described by a height function above the plane. In other words, they describe a <em>convex</em> shape with reflection symmetry at the z=0 plane.</p>
<p>Then the only thing you may have to modify is the lighting and the ratio of th... |
248,710 | <p>The organizers of a cycling competition know that about 8% of the racers use steroids. They decided to employ a test that will help them identify steroid-users. The following is known about the test: When a person uses steroids, the person will test positive 96% of the time; on the other hand, when a person does not... | Hagen von Eitzen | 39,174 | <p>We are given that $P(S)=0.08$ (hence $P(\neg S)=0.92$), $P(P|S)=0.96$ and $P(P|\neg S)=0.09$.
What we want to knwo is $P(S|P)$.</p>
<p>Note that $P(S\cap P)=P(S|P)\cdot P(P)$ as well as $P(S\cap P)=P(P|S)\cdot P(S)$, therefore
$$ P(S|P) = \frac{P(P|S)\cdot P(S)}{P(P)}.$$
Thus we first need $P(P)$, which we get from... |
2,305 | <p>I need an algorithm to produce all strings with the following property. Here capital letter refer to strings, and small letter refer to characters. $XY$ means the concatenation of string $X$ and $Y$.</p>
<p>Let $\Sigma = \{a_0, a_1,\ldots,a_n,a_0^{-1},a_1^{-1},\ldots,a_n^{-1}\}$ be the set of usable characters. Eve... | deinst | 943 | <p>Making explicit what is implicit in Qiaochu Yuan's comment, and demonstrating that <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.97.4244&rep=rep1&type=pdf" rel="nofollow">someone else's work</a> has failed to evade my eyes. (It is a neat article, read it.) I present this adaptation of Du... |
1,821,800 | <p>Consider the system of ODE in $\Bbb R^2 $ </p>
<p>$\dfrac{dY}{dt}=AY$ where $Y(0)=$ \begin{bmatrix} 0 \\ 1\end{bmatrix} $t>0$ </p>
<p>where $ A=$ \begin{bmatrix} -1 & 1 \\ 0 & -1\end{bmatrix}</p>
<p>and $Y(t)=$\begin{bmatrix} y_1(t) \\ y_2(t)\end{bmatrix}</p>
<p><strong>My try</strong>:
$dy_1(t)=-y_1(... | Arthur | 15,500 | <p>Note that if $\{x^2\} + \{x\} = 1$, then $x^2 + x$ is an integer. Solve the equation $x^2 + x = n$ for an arbitrary $n$, and see that if it has rational solutions, then those rationals must be integers, which means that $\{x^2\} + \{x\} = 0$.</p>
<p>To see that $x^2 + x$ must be an integer, note that for any $y$ we... |
439,918 | <p>I'm trying to find an example of a space that is Hausdorff and locally compact that is not second countable, but I'm stuck. I search an example on the book Counterexamples in Topology, but I can't find anything.<br>
Thank you for any help.</p>
| Asaf Karagila | 622 | <p><strong>Hint:</strong> Discrete spaces are locally compact.</p>
|
507,467 | <p>How do you factor $x^3 + x - 2$?</p>
<p>Hint: Write it as $(x^3-x^2+x^2-x+2x-2)$ to get $(x-1)(x^2+x+2)$</p>
<p>Note the factored form <a href="http://www.wolframalpha.com/input/?i=x%5E3+%2B+x+-+2" rel="nofollow noreferrer">here</a>. Thanks!</p>
| Mikasa | 8,581 | <p>Note that the summation of the coefficients is $0$:
$$+1+1+(-2)=0$$
so the polynomial has a factor like $(x-1)$.</p>
|
470,739 | <p>Assume $S$ and $T$ are diagonalizable maps on $\mathbb{R}^n$ such that $S\circ T$=$T \circ S$. Then $S$ and $T$ have a common eigenvector.</p>
<p>I already have proof, but I just need validation in one part.
My proof:
Let $F$ be an eigenvector of $T$. This means $\exists \; \lambda \in R$ such that $T(v)=\lambda v$... | Pete L. Clark | 299 | <p>Since I keep hinting that I want to see a certain answer, perhaps I had better just post it.</p>
<p>The OP has shown that for every <span class="math-container">$\lambda \in \mathbb{R}$</span>, the <span class="math-container">$\lambda$</span>-eigenspace <span class="math-container">$E_{\lambda}(T)$</span> is an <sp... |
1,774,670 | <p>Among many fascinating sides of mathematics, there is one that I praise, especially for didactic purposes : the parallels that can be drawn between some "Continuous" and "Discrete" concepts.</p>
<p>I am looking for examples bringing a help to a global understanding...</p>
<p>Disclaimer : Being d... | Artem | 29,547 | <p>My favorite one is about the discrete analogue of the wave equation. We all know how to solve the wave equation
$$
u_{tt}=\alpha^2u_{xx},\quad u(0,t)=u(1,t)=0,\quad u(x,0)=f(x),\,u_t(x,0)=g(x)
$$
with the separation of variables. However rigorously it requires the notion of Fourier series, convergence and the fact ... |
2,412,454 | <p>I was obviously not clear enough in my first question, so I will reformulate. I have the following equation
$$
A=\frac{B\sin 2\theta}{C+D\cos 2\theta}
$$
where $A,B,C,D$ are variables.
I need to solve or rewrite the equation to easily obtain $\theta$ (or $2\theta$), given known values for $A, B, C, D$.
Thanks for a... | trying | 309,917 | <p>A relation $R$ is said <em>in</em> a set $A$ when the $\operatorname{field} R\subseteq A$, where the $\operatorname{field} R=\operatorname{dom}R\cup\operatorname{range}R$. It is also said in this case that $R$ is a relation <em>between</em> elements of $A$. </p>
|
2,781,017 | <p>I known that $\sum a_i b_i \leq \sum a_i \sum b_i$ for $a_i$, $b_i > 0$. It seems this inequality will also hold true when $a_i$, $b_i \in (0,1)$. However, I am unable to find out if</p>
<p>$\sum \frac{a_i}{b_i} \leq \frac{\sum a_i}{\sum b_i}$ </p>
<p>holds true for $a_i$, $b_i \in (0,1)$.</p>
| Community | -1 | <p>It doesn't take long to find a counterexample.</p>
<p>$$\frac11+\frac11>\frac22.$$</p>
<p>Note that the restriction to $(0,1)$ is immaterial as $\dfrac ab=\dfrac{ca}{cb}.$</p>
|
623,190 | <p>What would be the formula, to determine a rectagles edges, when given the perimeter and space? for example, the rectagles space is 80, and the perimeter is 36, and the edge would be 8 and 10, but how do I find them.</p>
<p>I know that the formula for the perimeter would be
2x+2y=per, or 2(space/y)+2y=per
However I'... | Community | -1 | <p>By space, do you mean area?</p>
<p>So we know that $$2x + 2y = P$$ and also that $$xy = A.$$ This means that, as you pointed out, $$2\frac{A}{y} + 2y = P$$ and thus that $$A + y^2 - \frac{P}{2}y = 0$$ which we get by multiplying both sides by $y$. You can use the quadratic formula to solve for $y$ now, where you ge... |
312,878 | <p>Why is $\mathbb{Z} [\sqrt{24}] \ne \mathbb{Z} [\sqrt{6}]$, while $\mathbb{Q} (\sqrt{24}) = \mathbb{Q} (\sqrt{6})$ ?</p>
<p>(Just guessing, is there some implicit division operation taking $2 = \sqrt{4}$ out from under the $\sqrt{}$ which you can't do in the ring?)</p>
<p>Thanks. (I feel like I should apologize for... | Community | -1 | <p>We have</p>
<p>\begin{align*}
\mathbb{Z}[\sqrt{24}] &= \{a + b\sqrt{24} | a, b \in \mathbb{Z} \} \\
&= \{a + 2b\sqrt{6} | a, b \in \mathbb{Z} \} \\
&= \{a + b'\sqrt{6} | a, b' \in \mathbb{Z} \text{ with } b' \text { even}\}.
\end{align*}</p>
<p>which is clearly a proper subring of $\mathbb{Z}[\sqrt{6}... |
2,213,807 | <p>I was solving a problem to discover n and after I transformed the problem it gave me this equation:</p>
<p>\begin{equation*}
\left\lfloor{\frac{2}{3}\sqrt{10^{2n}-1}}\right\rfloor = \frac{2}{3}(10^{n}-1)
\end{equation*}</p>
<p>So I tried to simplify it by defining:
\begin{equation*}
k = 10^{n}-1
\end{e... | dxiv | 291,201 | <p>Hint: $10^n-1$ is a multiple of $9$, so $\frac{2}{3}k$ is an integer, then:</p>
<p>$$
\begin{align}
\left\lfloor{\frac{2}{3}\sqrt{k(k+2)}}\right\rfloor = \frac{2}{3}k \;\;&\iff\;\; \frac{2}{3}k \le \frac{2}{3}\sqrt{k(k+2)} \lt \frac{2}{3}k \,+\, 1 \;\; \\
&\iff\;\; \frac{4}{9}k^2 \le \frac{4}{9}(k^2+2k) \l... |
95,126 | <p>Consider the finite sum</p>
<pre><code>rs[x_, n_] := x/n Sum[n^2/(i + (n - i) x)^2, {i, 1, n}]
</code></pre>
<p>Is there a way to bring <em>Mathematica</em> to calculate the limit for <code>n -> ∞</code>?</p>
<p>I have tried <code>Limit[]</code> as well as <code>NLimit[]</code> without success.</p>
| J. M.'s persistent exhaustion | 50 | <p>This post tackles the convergence acceleration of the Riemann integral in the same spirit as Anton's answer, except that I use a slight variation of one of the algorithms presented. In particular, I'm using this as an excuse to present the <a href="http://dx.doi.org/10.1137/0510061" rel="noreferrer">van den Broeck-S... |
2,302,966 | <p>Translate the following English statements into predicate logic formulae. The domain is the set of integers. Use the following predicate symbols, function symbols and constant symbols.</p>
<ul>
<li>Prime(x) iff x is prime</li>
<li>Greater(x, y) iff x > y</li>
<li>Even(x) iff x is even</li>
<li>Equals(x,y) iff x=y</... | C. Falcon | 285,416 | <p>The answer is <strong>yes</strong>.</p>
<p><strong>Example.</strong> Let $G:=\mathrm{GL}(n,\mathbb{R})$, then $H:=O(n)$ is a compact subgroup of $G$ which is non-compact. Indeed, for all $k\geqslant 1$, $\textrm{diag}(k,1,\ldots,1)\in G$, whence $G$ is non-bounded and therefore non-compact. Besides, by definition $... |
2,302,966 | <p>Translate the following English statements into predicate logic formulae. The domain is the set of integers. Use the following predicate symbols, function symbols and constant symbols.</p>
<ul>
<li>Prime(x) iff x is prime</li>
<li>Greater(x, y) iff x > y</li>
<li>Even(x) iff x is even</li>
<li>Equals(x,y) iff x=y</... | CopyPasteIt | 432,081 | <p>The unit circle of the complex numbers (multiplicative group with $0$ removed) is a compact subgroup.</p>
<p>Similarly, $\{-1,+1\}$ is a compact subgroup of the nonzero real numbers.</p>
|
1,371,649 | <p>The question is:</p>
<blockquote>
<p>What does the following interation formula do?:
<span class="math-container">$$x_{k+1}=2x_k-cx_{k}^2.$$</span></p>
</blockquote>
<p>I tried to identify this with Newtons method. I.e. I tried to bring that into the form <span class="math-container">$x_{k+1}=x_k-\frac{f(x_0)}{f'(x_... | Community | -1 | <p>If the iterations converge, they converge to an <span class="math-container">$x$</span> such that</p>
<p><span class="math-container">$$x=2x-cx^2$$</span></p>
<p>and <span class="math-container">$x=0$</span> or <span class="math-container">$x=\dfrac1c$</span>.</p>
<p>Now the derivative of <span class="math-container... |
2,369,717 | <p>From Jaynes' probability theory: the logic of science, I found this:</p>
<blockquote>
<p><a href="https://i.stack.imgur.com/bnogp.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/bnogp.png" alt="enter image description here"></a></p>
</blockquote>
<p>$p$ here is the joint probability distributi... | Siong Thye Goh | 306,553 | <p>$\times$ means the regular multiplication between two real numbers, not the cartesian product.</p>
<p>The definition of independent is </p>
<p>$$P(X=x, Y=y) = P(X=x)P(Y=y)$$</p>
<p>You might want to understand the equation as</p>
<p>$$\int \int \rho(x,y) dxdy = \int f_X(x) dx \int f_Y(y) dy$$</p>
<p>Credit: Hag... |
829,390 | <p>In a tennis tournament, there are $10$ players. In the first round, $5$ groups(of 2 players) will be formed among them and elimination matches will be held among the two players in each group. In how many ways can pairings be done?</p>
<p>Answer is given as : $\frac{10!}{2^5\times5!}$</p>
<p>My solution :</p>
<p>... | Pavan Sangha | 154,686 | <p>If in your format you had match 1, match 2,...,match 5 and each was played at a different venue, then your answer would be correct. However this is not the case as you are only interested in sets of pairings. Suppose $A=\{1,2\} B=\{3,4\},...,D=\{9,10\}$ then one of the ways to obtain this set of pairings could be to... |
3,248,863 | <p>I want to calculate the operator norm of the operator <span class="math-container">$A: L^2[0,1] \to L^2[0,1]$</span> which is defined by <span class="math-container">$$(Af)(x):=i\int\limits_0^x f(t)\,dt-\frac{i}{2} \int\limits_0^1 f(t)\, dt$$</span></p>
<p>I've already shown that this operator is compact and selfad... | thehardyreader | 432,413 | <p>The eigenfunctions of the operator <span class="math-container">$A$</span> form an orthonormal system, therefore we can write:
<span class="math-container">$$Af = \sum\limits_{k\in\mathbb{Z}} \lambda_k (f,e_k)e_k$$</span> Where <span class="math-container">$\lambda_k = \frac{1}{(2k+1)\pi}$</span> are the eigenvalues... |
2,061,063 | <p>Let $X \subset C(\mathbb R;\mathbb R)$ be the space of all continuous functions $u: \mathbb R \to \mathbb R$ where </p>
<p>$$\lim_{x \to \pm \infty} u(x)=0$$</p>
<p>provided with the $\sup$-norm. Let $k \in L^1(\mathbb R)$, $u \in X$ and </p>
<p>$$(Ku)(x) := \int_\mathbb R k(x-y)u(y)\,dy, \,\,\,x \in \mathbb R.$$... | Olivier Moschetta | 369,174 | <p>Fix $f\in X$ and consider the mapping $\varphi:X\rightarrow X$ defined by
$$\varphi(u)=Ku+f$$
We apply the Banach-fixed point theorem on $\varphi$. To do so we need to check three conditions:</p>
<ul>
<li><p>$X$ is complete (enough to show that is closed in the Banach space of bounded continuous functions).</p></li... |
481,764 | <p>What type of symmetry does the function $y=\frac{1}{|x|}$ have?
Specify the intervals over which the function is increasing and the intervals where it is decreasing.</p>
| Eleven-Eleven | 61,030 | <p>Let $f(x)=y$</p>
<p>HINT: what does $f(-x)$ equal?</p>
<p>Let's look at the function, $f(x)=\frac{1}{x}$. Now, a function is even if:
$$f(-x)=f(x)$$
and a function is odd if:
$$f(-x)=-f(x)$$
We therefore can see that this function is an odd function, since;
$$f(-x)=\frac{1}{-x}=-\frac{1}{x}=-f(x)$$
Now, the abso... |
2,979,271 | <p>I have been able to calculate the integral of </p>
<p><span class="math-container">$$\int^\infty_\infty x^2e^{-x^2/2}$$</span></p>
<p>and there is a lot of information online about integration with even powers of <span class="math-container">$x$</span>.<br>
However I have been unable to calculate: </p>
<p><spa... | José Carlos Santos | 446,262 | <p>Do you mean <span class="math-container">$\int_{-\infty}^\infty x^3e^{-\frac{x^2}2}\,\mathrm dx$</span>? It is <span class="math-container">$0$</span>, since the function is an odd function and integrable (it is the product of a polynomial function with <span class="math-container">$e^{-\frac{x^2}2}$</span>).</p>
|
2,979,271 | <p>I have been able to calculate the integral of </p>
<p><span class="math-container">$$\int^\infty_\infty x^2e^{-x^2/2}$$</span></p>
<p>and there is a lot of information online about integration with even powers of <span class="math-container">$x$</span>.<br>
However I have been unable to calculate: </p>
<p><spa... | Dr. Sonnhard Graubner | 175,066 | <p>Substitute <span class="math-container">$$u=x^2$$</span> then we get <span class="math-container">$$\frac{1}{2}\int e^{-u/2}udu$$</span> and then use Integration by parts.</p>
|
4,004,978 | <blockquote>
<p>For all <span class="math-container">$a, b, c, d > 0$</span>, prove that
<span class="math-container">$$2\sqrt{a+b+c+d} ≥ \sqrt{a} + \sqrt{b} + \sqrt{c} + \sqrt{d}$$</span></p>
</blockquote>
<p>The idea would be to use AM-GM, but <span class="math-container">$\sqrt{a} + \sqrt{b} + \sqrt{c} + \sqrt{d}... | See Hai | 646,705 | <p>Alternative solution using Cauchy-Schwarz, which finishes the problem off immediately. By C-S, we have:
<span class="math-container">\begin{align}
& (a+b+c+d)(1+1+1+1) \geq (\sqrt{a}+ \sqrt{b}+ \sqrt{c}+ \sqrt{d} )^2 \\
& \Rightarrow 4(a+b+c+d) \geq (\sqrt{a}+ \sqrt{b}+ \sqrt{c}+ \sqrt{d} )^2 \\
& \Righ... |
1,929,445 | <blockquote>
<p>Is there a solution to the problem $$\left\{\begin{matrix}
y'=y+y^4\\
y(x_0)=y_0
\end{matrix}\right.$$
which is defined on $\mathbb{R}$? ($x_0,y_0$ might be any real numbers)</p>
</blockquote>
<p>It's easy to prove that for all $(x,y)\in\mathbb{R}^2$ there exists an open interval $I$ (with $x_0\in... | Lutz Lehmann | 115,115 | <p>This is a Bernoulli equation, and can thus be solved directly. Setting <span class="math-container">$u(x)=y(x)^{-3}+1$</span> results in the equation
<span class="math-container">$$
u'(x)=-3y(x)^{-4}y'(x)=-3u(x)\implies u(x)=u(x_0)e^{-3(x-x_0)}
$$</span>
so that
<span class="math-container">$$
y(x)=\frac{y(0)e^{x-x_... |
506,152 | <p>Is $$\frac{a+b}{c+d}<\frac{a}{c}+\frac{b}{d}$$
for $a,b,c,d>0$</p>
<p>If it is true, then can we generalize?</p>
<p>EDIT:typing mistake corrected.</p>
<p>EDIT, WILL JAGY. Apparently the <strong>real question</strong> is
Is $$\color{magenta}{\frac{a+b}{c+d}<\frac{a}{c}+\frac{b}{d}}$$
for $a,b,c,d>0,$ ... | Thomas Andrews | 7,933 | <p>If you consider them as slopes, then $(0,0)$, $(b,a)$, $(d,c)$ and $(b+d,a+c)$ form a parallelogram. So the slope of the line between $(0,0)$ and $(b+d,a+c)$ will be between the slopes of the lines between $(0,0)$ and $(b,a)$ and $(d,c)$. That means that $\frac{a+c}{b+d}$ will be between $\frac{a}{b}$ and $\frac{c}{... |
2,801,643 | <p>given $V$ be a vector space over $\mathbb{F}$. let $P:V\longrightarrow V$ be a function $\negthickspace$ as $V=U_{1}\oplus U_{2}$, and for every $v$, $v=u_{1}+u_{2}$ as $u_{1}\in U_{1}$ , $\!u_{2}\in U_{2}$. </p>
<p>$P(v)=p(u_{1}+u_{2})=u_{1}$. which means $P$ is a Projection.</p>
<p>I need to prove that p is a li... | Emilio Novati | 187,568 | <p>Hint.</p>
<p>The key fact is that, if $x=x_1+x_2$ with $x_1 \in U_1$ and $x_2 \in U_2$, and $y=y_1+y_2$ with $y_1 \in U_1$ and $y_2 \in U_2$ than we have:
$$
x+y= (x_1+x_2)+(y_1+y_2)=(x_1+y_1) + (x_2+y_2)
$$</p>
<p>with:
$$
x_1+y_1 \in U_1 \quad and \quad x_2+y_2 \in U_2
$$</p>
<p>Use this in the definition of $P... |
2,801,643 | <p>given $V$ be a vector space over $\mathbb{F}$. let $P:V\longrightarrow V$ be a function $\negthickspace$ as $V=U_{1}\oplus U_{2}$, and for every $v$, $v=u_{1}+u_{2}$ as $u_{1}\in U_{1}$ , $\!u_{2}\in U_{2}$. </p>
<p>$P(v)=p(u_{1}+u_{2})=u_{1}$. which means $P$ is a Projection.</p>
<p>I need to prove that p is a li... | Lennart Döppenschmitt | 564,574 | <p>Let's clean up the notation a little bit: Given a a vector space $V$ over $\mathbb{K}$ which decomposes as the direct sum $V = V_1 \oplus V_2$. Then every element $v \in V$ can be written as $v = v_1 + v_2$ with $v_i \in V_i$.</p>
<p>The projection $P: V \rightarrow V, v \mapsto v_1$ is a linear transformation beca... |
3,063,053 | <p>I'm a Calculus I student and my teacher has given me a set of problems to solve with L'Hoptial's rule. Most of them have been pretty easy, but this one has me stumped. <br /></p>
<p><span class="math-container">$$\lim\limits_{x\to \infty} \frac{x}{\sqrt{x^2 + 1}}$$</span> </p>
<p>You'll notice that using L'Hopital... | kkc | 630,558 | <p>When computing the limit of rational functions, as is the case for <span class="math-container">$$\lim_{x \rightarrow \infty} \frac{x}{\sqrt{x^2 +1}},$$</span> you want to divide the top and bottom by the highest degree in the denominator, which in this case is <span class="math-container">$x$</span>. Since <span cl... |
2,218,924 | <p>$ \displaystyle \lim_{n\to \infty} \sum_{k=1}^n \frac{k^4}{n^4}=$ ?</p>
<p>I found it difficult to tranform it into the integral form by the definition of Riemann sum, which is a way to solve similar problems.</p>
| Bernard | 202,857 | <p><strong>Hint:</strong></p>
<p>$$\sum_{k=1}^n \frac{k^4}{n^4}=\frac1{n^4}\sum_{k=1}^n k^4$$
and you can use <em>Faulhaber's formula</em> to get the value of
$S_4(n)=\sum_{k=1}^n k^4$ in function of $$S_3(n)=\dfrac{n^2(n+1)^2}4,\quad S_2(n)=\dfrac{n(n+1)(2n+1)}6,\quad S_1(n)=\dfrac{n(n+1)}2.$$
You obtain a polynomial... |
2,837,172 | <p>In complex analysis, sometimes we need to use some theorems which are results of measure theory. However, I know very very little about measure theory. So</p>
<blockquote>
<p>What are some very basic results of measure theory on complex functions/complex plane/complex calculus?</p>
</blockquote>
<p>I expect the ... | Homieomorphism | 553,656 | <p>I would suggest you take a look at Folland : Real Analysis and Modern Techniques. The chapter 0 can be ommited and simply used as reference when you forget some basic analysis results. Chapter 1 is a bit harsh at the beginning so maybe skim through it to have a general idea of what is a measure but the most interest... |
1,780,797 | <p>According to <a href="https://en.wikipedia.org/wiki/Null_set#Lebesgue_measure" rel="nofollow noreferrer">Wikipedia</a>, the straight line <span class="math-container">$\mathbb{R}$</span> is a null set in <span class="math-container">$\mathbb{R}^2$</span>.</p>
<p>That means, the line <span class="math-container">$\ma... | Jack D'Aurizio | 44,121 | <p>Let $f(t)$ be the PDF of $X$ and $g(t)$ be the PDF of $Y$.
$$D_{KL}(P_X\parallel P_{X+Y}) = \int_{-\infty}^{+\infty}f(x)\log\frac{f(x)}{(f*g)(x)}\,dx$$
does not admit any obvious simplification, but the term </p>
<p>$$\log\frac{f(x)}{(f*g)(x)}=\log\frac{\int_{-\infty}^{+\infty} f(t)\,\delta(x-t)\,dt}{\int_{-\infty}... |
217,514 | <p>Given $A$, $B$ are bounded subsets of $\Bbb R$.
Prove </p>
<ol>
<li>$A\cup B$ is bounded. </li>
<li>$\sup(A \cup B) =\sup\{\sup A, \sup B\}$.</li>
</ol>
<p>Can anyone help with this proof?</p>
| mohamez | 34,920 | <p>for $1$ use the fact that $x\in A\cup B \Leftrightarrow x\in A$ or $x\in B$ (notice that $SupA,\space SupB$ exists since $A,\space B$ are bounded) and for $2$ use the least upper bound property. that if $SupA = M \Leftrightarrow \forall x\in A,\space \exists M\in \mathbb{R}$ such that $x\leq M$ and $\forall \epsilon... |
217,514 | <p>Given $A$, $B$ are bounded subsets of $\Bbb R$.
Prove </p>
<ol>
<li>$A\cup B$ is bounded. </li>
<li>$\sup(A \cup B) =\sup\{\sup A, \sup B\}$.</li>
</ol>
<p>Can anyone help with this proof?</p>
| Brian M. Scott | 12,042 | <p>Without loss of generality assume that $\sup A\le\sup B$, so that $\sup\{\sup A,\sup B\}=\sup B$, and you simply want to show that $\sup(A\cup B)=\sup B$. Clearly $\sup(A\cup B)\ge\sup B$, so it suffices to show that $\sup(A\cup B)\le\sup B$.</p>
<p>To show that $\sup(A\cup B)\le\sup B$, just prove that $\sup B$ is... |
2,590,165 | <p>How to show $f(x)$=$\frac{1}{1+x^2}$ is uniform continuous on $\Bbb R$. </p>
<p>Although, of course for any interval $[a,b]$, this function is continuous and bounded, therefore also uniformly continuous. Following <strong>Continuous Extension Theorem</strong> it is uniformly continuous on any $(a,b)$. Therefore pr... | user | 505,767 | <p>Note that</p>
<p>$$\frac{e^{xB}}{e^{xA}} \gt \frac{xA-1}{xB-1}\iff xB-xA >\log(1-xA)-\log(1-xB)$$</p>
<p>$$\iff \log(1-xB)+xB>\log(1-xA)+xA$$</p>
<p>which is false, indeed</p>
<p>$$f(y)=\log(1-y)+y\implies f'(y)=\frac{-1}{1-y}+1=\frac{-y}{1-y}<0 \quad y\in(0,1)$$</p>
|
452,803 | <p>Test the convergence of improper integrals :</p>
<p>$$\int_1^2{\sqrt x\over \log x}dx$$</p>
<p>I basically have no idea how to approach a problem in which log appears. Need some hint on solving this type of problems.</p>
| Random Variable | 16,033 | <p>$$ \lim_{x \to 1^{+}} (x-1) \frac{\sqrt{x}}{\ln x} = \lim_{x \to 1^{+}} \frac{\sqrt{x} + (x-1) \frac{1}{2 \sqrt{x}}}{\frac{1}{x}} = 1 $$</p>
<p>The integrand behaves like $\frac{1}{x-1}$ near $x=1$ and thus $ \displaystyle\int_1^2{\sqrt x\over \log x}dx$ diverges.</p>
|
378,953 | <p><strong>Problem:</strong> Give an example of a permutation of the first $n$ natural numbers from which it is impossible to get to the standard permutation $1,2,\ldots,n$ after less than $n-1$ transposition operations (i.e switching the place of two elements).</p>
<p><strong>My attempt</strong></p>
<p>Suppose we ha... | Ted | 15,012 | <p>Yes, this is correct. As you observed, the key fact is that hitting a permutation with a transposition $(ij)$ always either decreases or increases the number of cycles by exactly 1. If the original permutation had $i$ and $j$ in the same cycle, then it splits them (hence increases the number of cycles by 1); if th... |
156,376 | <p>I understand that when we are doing indefinite integrals on the real line, we have $\int f(x) dx = g(x) + C$, where $C$ is some constant of integration. </p>
<p>If I do an integral from $\int f(x) dx$ on $[0,x]$, then is this considered a definite integral? Can I just leave out the constant of integration now? I am... | cuabanana | 64,547 | <p>Yes, your function is a definite integral, because it is evaluated over a certain interval. Although the constant is strictly not necessary, because it will be subtracted when the integral is evaluated, it is good practice to keep the constant of integration. If you want to be consistent, rename the variable in the ... |
3,832,684 | <p>Does the following inequality hold?
<span class="math-container">$$\sqrt {x-z} \geq \sqrt x -\sqrt{z} \ , $$</span>
for all <span class="math-container">$x \geq z \geq 0$</span>.</p>
<p>My justification
<span class="math-container">\begin{equation}
z \leq x \Rightarrow \\ \sqrt z \leq \sqrt {x} \Rightarrow \\ 2\sqr... | poetasis | 546,655 | <p><span class="math-container">\begin{equation}
\qquad\sqrt {x-z} \ge \sqrt x -\sqrt{z}\\
\implies (\sqrt {x-z})^2 \geq (\sqrt x -\sqrt{z})^2\\
\implies x-z\ge x-2\sqrt{xz}+z\\
\implies x-z - x-z\ge-2\sqrt{xz}\\
\implies -2z\ge -2\sqrt{xz}\\
\implies -z\ge -\sqrt{xz}\\
\text{ subtracting both sides from both sides rev... |
4,515,517 | <p>Suppose that <span class="math-container">$E$</span> is a measruable set and <span class="math-container">$f: E \rightarrow [0, \infty]$</span> is a non-negative function with <span class="math-container">$\int_E f(x)^n dx = \int_E f(x) dx < \infty$</span> for all positive integers <span class="math-container">$n... | Adayah | 149,178 | <p>First of all, I think it is valuable that you try to carry out Shashi's technical approach from the comments. Having said that, there is a nice trick that solves the problem: let</p>
<p><span class="math-container">$$P(y) = y^2 (y-1)^2 = y^4 - 2y^3 + y^2.$$</span></p>
<p>By assumption</p>
<p><span class="math-contai... |
2,877,085 | <p>I think that there could be used Abel and Dirichlet method, but I have no idea how</p>
<p>$$ \sum_{n=1}^{\infty} (-1)^n\frac{3n-2}{n+1}\frac{1}{n^{1/2}} .$$</p>
| Angina Seng | 436,618 | <p>The series
$$\sum_{n=1}^\infty\frac{3(-1)^n}{n^{1/2}}$$
is convergent by Leibniz. The difference from the original series
is
$$\sum_{n=1}^\infty (-1)^n\left(\frac{3n-2}{(n+1) n^{1/2}}-\frac{3}{n^{1/2}}\right)$$
which is aboslutely convergent, since the terms are $O(n^{-3/2})$.</p>
|
1,661,244 | <p>If $R$ is a comutative ring with identity ring and $K$ is an ideal from it, let $R'=R/K$ and $I$ an ideal of $R$ satisfy $K\subseteq I$ and $I'$ is the coresponding ideal of $R'$ (we knew that correspondence theorem gives a certain one-to-one corespondence between the set of ideals of $R$ containing $K$ and the set ... | MooS | 211,913 | <p>I do not know why you want to replace 'prime' by 'principal', since these properties do not really relate, but here is an example:</p>
<p>$R=k[x,y], K=(x), I=(x,y)$. $I$ is not principal but $I'=I/K=(y)$ is principal in $R'=R/K=k[y]$.</p>
|
750,751 | <p>if $V$ is a finite-dimensional vector space and $t \in \mathcal L (V,V) $is such that $t^2 = id_V$ prove that the sum of eigenvalues of t is an integer.</p>
<p>I started the prove as such:</p>
<p>Let $\lambda_1 ,...,\lambda_n $ be eigenvalues of $t$. </p>
<p>So $\lambda_1^2 ,... \lambda_n^2$ will be the eigenvalu... | Pedro | 23,350 | <p>You know $X^2-1=(X-1)(X+1)$ annihilates $t$. What can be the possible eigenvalues for $t$?</p>
|
3,041,632 | <p><span class="math-container">$X_n=4X_{n-1}+5$</span></p>
<p>How come the solution of this recurrence is this? </p>
<p><span class="math-container">$X_n=\frac834^n+\frac53$</span></p>
<p>I also have that <span class="math-container">$X_0=1$</span>.</p>
<p>I am using telescoping method and I am trying to solve it ... | Asit Srivastava | 567,604 | <p>This is a difference equation and it can be solved using Z-Transform. Take Z-transform of both sides of the equation and then use the initial condition. Ultimately, you will get Z-transform of X and then take its inverse z-transform to get the solution.</p>
|
3,041,632 | <p><span class="math-container">$X_n=4X_{n-1}+5$</span></p>
<p>How come the solution of this recurrence is this? </p>
<p><span class="math-container">$X_n=\frac834^n+\frac53$</span></p>
<p>I also have that <span class="math-container">$X_0=1$</span>.</p>
<p>I am using telescoping method and I am trying to solve it ... | lab bhattacharjee | 33,337 | <p>Let <span class="math-container">$x_m=y_m+a\implies y_0=x_0-a=1-a$</span></p>
<p><span class="math-container">$$5=x_n-4x_{n-1}=y_n+a-4(y_{m-1}+a)=y_n-4y_{n-1}-3a$$</span></p>
<p>Set <span class="math-container">$-3a=5\iff a=?$</span> so that <span class="math-container">$y_n=4y_{n-1}=\cdots=4^ry_{n-r},0\le r\le n$... |
3,153,821 | <p>I'm trying to analyse a game of Mastermind and am having trouble quantifying the amount of possible game states. I know that a code has <span class="math-container">$\text{# of colors}^{\text{# of pegs per guess}}$</span> combinations (in my case that would be <span class="math-container">$6^4=1296$</span>). However... | Especially Lime | 341,019 | <p>Your formula going from rows to the full board is incorrect, and should be <span class="math-container">$\#\text{combinations per row}^{\#\text{rows}}$</span>, giving <span class="math-container">$1296^{11}$</span> which is much less. Substituting in the formula for combinations per row, this is just <span class="ma... |
2,933,572 | <p>Suppose <span class="math-container">$A = 1/2^{100\log(n)}$</span>, and <span class="math-container">$B = e^{-100\log(2) \log(n)}$</span>.</p>
<p>I'm required to prove that <span class="math-container">$A$</span> and <span class="math-container">$B$</span> are equal, how should I prove this? I tried applying some r... | mechanodroid | 144,766 | <p>Let <span class="math-container">$(x_n)_n$</span> be a sequence in <span class="math-container">$C$</span> which converges to <span class="math-container">$x \in \mathbb{R}$</span>. We have
<span class="math-container">$$d(x, C) \le d(x, x_n) \xrightarrow{n\to\infty} 0$$</span></p>
<p>so <span class="math-container... |
2,120,194 | <blockquote>
<p>Let $K_1$ and $K_2$ be two disjoint compact sets in a metric space $(X,d).$ Show that $$a = \inf_{x_1 \in K_1, x_2 \in K_2} d(x_1, x_2) > 0.$$
Moreover, show that there are $x \in K_1$ and $y \in K_2$ such that $a = d(x,y)$.</p>
</blockquote>
<p>For the first part, suppose to the contrary that $... | Ken Duna | 318,831 | <p>You have the right idea. The ideal $(x^2 - 2) \subseteq \mathbb{Q}[x]$ is maximal and so $R = \frac{\mathbb{Q}[x]}{(x^2-2)}$ is a field. </p>
<p>Note that $R \cong \{ a + b \sqrt{2} \ | \ a,b \in \mathbb{Q} \}$ and $\mathbb{Q} \subseteq R$.</p>
<p>Suppose that $\sqrt{3} \in R$. Then there exist $a,b \in \mathbb{Q}... |
2,795,777 | <p>I encountered this problem in one of my linear algebra homeworks (Linear Algebra with Applications 5th Ed 1.3.44):</p>
<p>Consider a $n \times m$ matrix $A$, such that $n > m$. Show there is a vector $b$ in $\mathbb{R}^{n}$ such that the system $Ax=b$ is inconsistent.</p>
<p>I have a strong intuition as to why ... | José Carlos Santos | 446,262 | <p><strong>Hint:</strong> What is $A.v$ if $v=(1,10,10^2,\ldots,10^9)$?</p>
|
250,119 | <p>I'd like to show that if a set $X$ is Dedekind finite then is is finite if we assume $(AC)_{\aleph_0}$. As set $X$ is called Dedekind finite if the following equivalent conditions are satisfied: (a) there is no injection $\omega \hookrightarrow X$ (b) every injection $X \to X$ is also a surjection.</p>
<p>Countable... | Matt E | 221 | <p>The Lefschetz principle can be understood in scheme theoretic terms in the following way:</p>
<p>suppose that $X \to S$ is a scheme over a base $S$ (possibly with extra data) which is fppf over $S$. Then we may descend $X$ to $X_0 \to S_0$ where $S_0$ is finite type over $\mathbb Z$. (Here "descend" means that th... |
5,586 | <p>I'm in my last year of highschool. And I'm aiming for a perfect grade in maths. The problem is that this year is the hardest year of maths I have ever faced in my entire life. Especially derivation and limits as its the first time I am studying it. Here are the lessons that are required to study for the first semest... | Joonas Ilmavirta | 2,074 | <p>The best way to ensure a good grade is to make sure you <em>deeply understand</em> the topics you are supposed to learn.
It is of course important to remember the routine solution methods, but you should also be able to tell intuitively and at a glance <em>why</em> these methods work and <em>where</em> any given met... |
5,586 | <p>I'm in my last year of highschool. And I'm aiming for a perfect grade in maths. The problem is that this year is the hardest year of maths I have ever faced in my entire life. Especially derivation and limits as its the first time I am studying it. Here are the lessons that are required to study for the first semest... | Jasper | 1,147 | <p>The advice that JPBurke and Joonas Ilmavirta have given is excellent.</p>
<p>If you want to be perfect, you need to check your work (as explained at the end of this answer). You also need to know that some problems do not have answers -- and that the correct answer may be to point out why. Furthermore, real-world... |
5,586 | <p>I'm in my last year of highschool. And I'm aiming for a perfect grade in maths. The problem is that this year is the hardest year of maths I have ever faced in my entire life. Especially derivation and limits as its the first time I am studying it. Here are the lessons that are required to study for the first semest... | guest | 8,441 | <p>The best way to do well in math is to solve problems. (Not to deeeeply understand things.) Deeply understanding things is fine. Even beneficial. but skill in problems is more important.</p>
<p>Per the actual question: yes, Schaum's Outlines. Perfect for what you want which is to drill the A.</p>
<p>Oh...and ... |
4,498,801 | <p>I am trying to deeply understand the similarities between these two theorems; the first being a generalization of the second.</p>
<blockquote>
<p><strong>Theorem 16.13.</strong> If <span class="math-container">$f$</span> is nonnegative, then
<span class="math-container">$$
\int_{\Omega} f(T \omega) \mu(d \omega)=\in... | Ruy | 728,080 | <p>A lot has been said about this question already, but a new perspective has just occured to me that might be useful.</p>
<p>First of all, let us consider a context in which we have two measure spaces <span class="math-container">$(X, \mu )$</span> and <span class="math-container">$(Y, \nu )$</span>, and a
measura... |
3,059,695 | <p>Let <span class="math-container">$A$</span> be a subset of a compact topological space such that every point of <span class="math-container">$A$</span> is an isolated point of <span class="math-container">$A$</span>. Is <span class="math-container">$A$</span> necessarily finite?</p>
| Henno Brandsma | 4,280 | <p>No, it can be very large. But if <span class="math-container">$A$</span> is also closed, it is compact and then it must be finite. </p>
<p>E.g. <span class="math-container">$[0,1]^{\mathbb{R}}$</span> is compact in the product topology, but the set <span class="math-container">$A$</span> of all functions <span clas... |
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