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 |
|---|---|---|---|---|
2,829,121 | <p>I get how to derive the ellipse equation, but I'm struggling to understand what it means intuitively. </p>
<p>You see, a circle equation can be understood very intuitively. The circle equation models how the radius of the circle can be represented using the Pythagorean theorem. But I don't understand what the ellip... | Blue | 409 | <p>I mentioned in my <a href="https://math.stackexchange.com/a/2829283/409">previous answer</a> that the "foci and string" form of the ellipse equation is rarely seen "except as the point of departure on an algebraic journey to the 'standard' form". I want to elaborate a little on that "algebraic journey".</p>
<p>Typi... |
1,609,443 | <p>So my textbook states that it is $(-\infty,0)$ that is concave down, but why not $(-\infty,0]$? Could someone help?</p>
| John Steinbeck | 304,183 | <p>Well, @User12345 if textbooks do that, they are certainly wrong.
$f:x\mapsto -x^4$ is certainly concave on $\mathbb{R}$ but do not satisfy your definition.</p>
<p>@CoolKid, I don't know the definition you have of a concave function, but for every definition I would agree with, I consider you are right and that $x\m... |
1,701,761 | <p><a href="https://i.stack.imgur.com/Qletj.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Qletj.png" alt="Trapezium "></a> </p>
<blockquote>
<p>Prove that ABED is a parallelogram </p>
</blockquote>
<p><strong>Given:</strong></p>
<ol>
<li><p>ABCD is a trapezium</p></li>
<li><p>F and G are the m... | Intelligenti pauca | 255,730 | <p>HINT: draw $AE$ and show that triangles $ABE$ and $ADE$ are congruent.</p>
|
171,697 | <blockquote>
<p>Let $f \in C^1(\mathbb R, \mathbb R)$ and suppose
$$ \tag{H}
\vert f(x) \vert\le \frac{1}{2}\vert x \vert + 3, \quad \forall x \in \mathbb R.
$$</p>
<p>Then, every solution of
$$
x'(t)+x(t)+f(x(t))=0, \quad t \in \mathbb R
$$
is bounded on $[0,+\infty]$. </p>
</blockquote>
<p>First of ... | AlbertH | 11,066 | <p>By assumptions
$$
x'(t) = -x(t) -f(x(t)) \leq -x(t) + \frac 1 2 \lvert x(t) \rvert + 3
$$
Let's suppose there is a point $t_0$ such that
$$
x(t_0) > \max\{x(0), 6\}
$$</p>
<p>Since $x(t_0)$ is positive the following relation is satisfied
$$
x'(t_0) \leq -x(t_0) + \frac 1 2 \lvert x(t_0) \rvert + 3 = -\frac 1 2 ... |
716,856 | <p>I have the following equation:</p>
<p>\begin{align*}
\frac{d^2\theta}{dt}=\alpha(\theta-1)+\beta(\theta-1)^3-\gamma\frac{d\theta}{dt} \tag{1}
\end{align*}</p>
<p>Where $\alpha, \beta, \gamma \in \mathbb{R}$.</p>
<p>This is my solution in attempting to convert (1) into a system of first order ODE's:</p>
<p>\begin... | Amzoti | 38,839 | <p>We are given:</p>
<p>\begin{align*}
\frac{d^2\theta}{dt}=\alpha(\theta-1)+\beta(\theta-1)^3-\gamma\frac{d\theta}{dt} \tag{1}
\end{align*}</p>
<p>Where $\alpha, \beta, \gamma \in \mathbb{R}$.</p>
<p>We can proceed as follows:</p>
<ul>
<li>$x_1 = \theta \implies x_1' = \theta' = x_2$</li>
<li>$x_2 = \theta' \impli... |
67,039 | <p>I just noticed that <code>Exit</code> and <code>Quit</code> can work without brackets i.e. a single</p>
<pre><code>Exit
</code></pre>
<p>or</p>
<pre><code>Quit
</code></pre>
<p>will quit the kernel. Quite surprising!</p>
<p>This isn't mentioned in the document of them. Is there a list for functions with this fe... | shelure21 | 62,528 | <pre><code>Here
</code></pre>
<p>Could also work without brackets.</p>
|
1,332,433 | <blockquote>
<p>Let <span class="math-container">$f:\mathbb R\to \mathbb R$</span> be such that for every sequence <span class="math-container">$(a_n)$</span> of real numbers,</p>
<p><span class="math-container">$\sum a_n$</span> converges <span class="math-container">$\implies \sum f(a_n)$</span> converges</p>
<p>Prov... | Justthisguy | 81,821 | <p>Here is a proof in the case that $f$ is differentiable at $0$.</p>
<p>Let $f(x) = rx + \epsilon(x)$, where $\lim_{x \to 0} {\epsilon(x)/x}=0$. I claim $\epsilon = 0$ in a a neighborhood of 0.
Suppose not. Then, up to extracting subsequences and making a few sign choices that are without loss of generality, we may a... |
50,441 | <p>here is the silly question of mine,</p>
<p>if <code>mathematica</code> takes <code>3</code> seconds to evaluate the function <code>f[3]</code>
then I plug the this value <code>f[3]</code> into a new function <code>g[f[3],f[3]]</code>, Let's ignore the evaluation time for the new function <code>g</code>, how long do... | Apple | 10,193 | <pre><code>Clear["Global`*"]
ClearSystemCache[];
$RecursionLimit = Infinity;
fibonacci[1] = 1; fibonacci[2] = 1;
fibonacci[i_] := fibonacci[i] = fibonacci[i - 1] + fibonacci[i - 2];
fibonacci[20000]; // AbsoluteTiming
fibonacci[20000]; // AbsoluteTiming
(*{0.135008, Null}*)
(*{0., Null}*)
</code></pre>
|
33,778 | <p>Let $X$ and $Y$ be independent exponential variables with rates $\alpha$ and $\beta$, respectively. Find the CDF of $X/Y$.</p>
<p>I tried out the problem, and wanted to check to see if my answer of: $\frac{\alpha}{ \beta/t + \alpha}$ is correct, where $t$ is the time, which we need in our final answer since we need... | Michael Hardy | 11,667 | <p>A different method may be of interest:
<span class="math-container">\begin{align}
\Pr(X/Y \le t) = {} & \Pr(X\le tY) \\[8pt]
= {} & \iint\limits_{(x,y)\,:\,x\,\le\, ty} e^{-\alpha x} e^{-\beta y} \big(\alpha\beta\,d(x,y)\big) \\[8pt]
= {} & \int_0^\infty \left( \int_0^{ty} e^{-(\alpha x+\beta y)} (\alpha... |
495,487 | <p>Show that the difference of the number of cycles of even and odd permutations is $(-1)^n (n-2)!$, <strong>using a bijective mapping</strong> (combinatorial proof).</p>
<p>Suppose to convert a permutation from odd to even we always flip 1 and 2. Then the difference can be easily calculated.</p>
<p>Thus, if I can fi... | Marko Riedel | 44,883 | <p>Here is an answer using generating functions that may interest you. Recall that the sign $\sigma(\pi)$ of a permutation $\pi$ is given by
$$\sigma(\pi) = \prod_{c\in\pi} (-1)^{|c|-1}$$
where the product ranges over the cycles $c$ from the disjoint cycle composition of $\pi$.</p>
<p>It follows that the combinatorial... |
1,938,353 | <p>A detailed solution will be helpful. Given that $a_1=3$ and $a_{n+1}=3^{a_n}$, find the remainder when $a_{2004}$ is divided by 100.</p>
| Jack D'Aurizio | 44,121 | <p>By the CRT, in order to find $a_{2004}\pmod{100}$ it is enough to find $a_{2004}\pmod{8}$ and $a_{2004}\pmod{25}$. Now $a_n=3^{a_{n-1}}\pmod{8}$ just depends on $a_{n-1}\pmod{4}$, and $a_{n-1}=3^{a_{n-2}}\pmod{4}$ just depends on $a_{n-2}$ being even or odd. Since $a_{2002}$ is clearly odd, $a_{2003}\equiv 3\pmod{4}... |
194,985 | <p>How does one show that $$\lim_{n \rightarrow \infty}\int_{0}^{1}\frac{x^{n}}{1 + x^{n}}\, dx = 0?$$ My idea is to evaluate the inner integral, but I can't seem to be able to do that.</p>
| Flanders | 39,649 | <p>For all $x\in [0,1]$ you have $$ \frac{x^n}{1+x^n} \leq x^n $$ and hence</p>
<p>$$ 0\leq\int_{0}^{1}\frac{x^{n}}{1 + x^{n}}\, dx \leq \int_{0}^{1}x^{n}\, dx = \frac{1}{n+1}\rightarrow 0 \quad (n\rightarrow \infty).$$</p>
|
96,518 | <p>Some time ago I had a physics test where I had the following integral: $\int y'' \ \mathrm{d}y$. The idea is that I had a differential equation, and I had acceleration (that is, $y''$) given as a function of position ($y$). The integral was actually equal to something else, but that's not the point. I neede... | Pedro | 23,350 | <p>I'm doubtfull about what $y''$ and $dy$ stand for in your problem. If you have $y = y(x)$ then clearly $$\int y'' dx = y'+C$$ But you're integrating with respect to $dy = d\{y(x)\} = y'(x) dx$ assuming $y(x)$ has a continuous derivative. So you finally have.</p>
<p>$$\int y''(x) y... |
1,006,080 | <p>How to solve:
$$\left|\frac{x+4}{ax+2}\right| > \frac{1}{x}$$</p>
<p>What I have done:<br><br></p>
<p>I) $x < 0$:</p>
<p>Obviously this part of the inequation is $x\in(-\infty, 0), x\neq \frac{-2}{a}$<br><br></p>
<p>II) $x > 0$:</p>
<p>$$\left|\frac{x+4}{ax+2}\right| > \frac1x$$
$$\frac{|x+4|}{|ax+2... | b.sahu | 60,209 | <p>The technique which you have used to accelerate the convergence of the function for "PI" , comes from Euler-Mclaurin Formula (see Wikipedia).This formula was developed around the year 1735 AD.....Archimedes method of finding approximations of "PI", was developed about 2000 years before the Euler-Maclaurin formula. <... |
176,192 | <p>I am trying to solve a system of differential equations as follows:</p>
<pre><code>L1 = 5.0;
L2 = 0.43;
d = 0.0056;
g = 9.8;
h1 = 5.0;
h2 = 0.43;
A1 = 0.00001;
A2 = 0.001;
p0 = 100000;
v0 = 0.001;
NDSolve[{P[t] -
1*1000*10*
L2 == (1000/(2*0.0056)*y[t]^2*
L2*(0.316*(1000*y[t]*0.0056/0.00089))^(-0.25)) +
500*01.... | halirutan | 187 | <p>I'm concentrating on the calculation of <code>samplemanycyclesper5years</code> and <code>samplecycledistributions</code>. For the first one, you select 1826 samples randomly and calculate the total. This is done <code>10^7</code> times. We can pack the random total into a compiled function that chooses 1826 random i... |
123,605 | <p>I have an equality:
$$\frac {d_L}{\phi t_1^2}=\frac {d_0}{\phi (t_1 - \Delta t)^2}$$
I can manually extract $d_L - d_0$ from this equation by expanding and re-arranging. Is there any way to do this in Mathematica? Ideally, I'd like <strong>Solve</strong> to work like this:</p>
<p>Solve$[\frac {d_L}{\phi t_1^2}==\... | march | 29,734 | <p>I usually do this by replacing one of the variables in the expression with the difference, then solve, then re-replace.</p>
<pre><code>Equal @@ Solve[eq /. d0 -> dl - dl0, dl0][[1, 1]] /. dl0 -> dl - d0
(* -d0 + dl == dl (2 t1 Δt - Δt^2)/t1^2 *)
</code></pre>
<p>In terms of <code>dl - d0</code> and <code>d0<... |
211,052 | <p>I want to modify a list of table row by adding an extra column. For that I <a href="https://reference.wolfram.com/language/ref/Map.html" rel="nofollow noreferrer"><code>Map</code></a> the data with a pure function that evluate the new column value from exisiting one and reconstruct a new list from that and the init... | MarcoB | 27,951 | <p>Two options presented themselves to me, using <code>Apply</code> or using replacements:</p>
<pre><code>{#1, #2, #3, #1 > 2 b, #4} & @@@ data
data /. {x_, y_, z_, p_} :> {x, y, z, x > 2 b, p}
</code></pre>
<p>Both reproduce your results.</p>
|
503,808 | <p>Let $\mathbb{K} = \mathbb{Q}[\sqrt[3]{5}] \ $, and let $\mathbb{L}$ be the normal closure of $\mathbb{K}$. </p>
<p>Let $\mathbb{O}_{\mathbb{K}} \ $ be the integral closure of $\mathbb{Z}$ in $K$ and $\mathbb{O}_{\mathbb{L}} \ $ be the integral closure of $\mathbb{Z}$ in $\mathbb{L}$. I want to find the factorizatio... | Lubin | 17,760 | <p>I tend to work on such things on a fairly mindless and <em>ad hoc</em> basis. Let me show you my approach just for the case $p=7$:</p>
<p>The extension $\mathbb K\supset\mathbb Q$ is of degree $3$, and in the downstairs residue field $\mathbb F_7$, $5$ is a generator of the multiplicative group so that $\lambda=5^{... |
1,179,498 | <p>Is the following sequence convergent in <img src="https://i.stack.imgur.com/wtYTG.png" alt="enter image description here">?(Metric space is the normal, euclidean space)
<img src="https://i.stack.imgur.com/uqGFl.png" alt="enter image description here"></p>
| quid | 85,306 | <p>Show: </p>
<ul>
<li><p>if the additive group contains an element of order $2$ then the multiplicative does not. </p></li>
<li><p>if the additive group contains no element of order $2$ then the multiplicative does. </p></li>
</ul>
|
3,164,343 | <p>Machine generates one random integer in range <span class="math-container">${[0;40)}$</span> on every spin.<br>
You should choose 5 numbers in that range. </p>
<p>Then the machine will spit out <strong>5</strong> numbers (<em>numbers are independent of each other</em>). </p>
<p><strong>what is the probability t... | SlipEternal | 156,808 | <p>If the machine is choosing with replacement, but the person is choosing five distinct numbers, and if success means two distinct numbers the person chose are picked by the machine, then the problem is much more difficult:</p>
<p>Probability machine chooses five distinct numbers:
<span class="math-container">$$\dfra... |
3,762,306 | <p>I have the equation <span class="math-container">$$(e^x-1)-k\arctan(x) = 0$$</span> where <span class="math-container">$0<k \leq \frac 2\pi$</span> and I was wondering how I would go about starting to determine the amount of real roots of this equation. So far I have just manipulated the equation to get different... | saulspatz | 235,128 | <p>Let <span class="math-container">$f(x)=e^x-1-k\arctan(x)$</span>. We have <span class="math-container">$f(0)=0.$</span> If <span class="math-container">$k=0$</span>, obviously <span class="math-container">$x=0$</span> is the only solution.</p>
<p>If <span class="math-container">$k\neq0$</span>, then <span class="m... |
1,652,381 | <p>Suppose $f$ derivable on $\mathbb R$ and that $\lim_{x\to a}f'(x)=\ell$. Show that $$\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}=\ell.$$</p>
<p>The proof of my course goes like this:</p>
<p>By mean value theorem, there is $y_h\in ]x,x+h[$ s.t. $$f(x+h)-f(x)=f'(y_h)h$$ and thus $$\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}=\lim_{h\t... | yo' | 43,247 | <p>$40$, because the sequence is $n+7$ and $n^2+10n+16$ interleaved.</p>
<p>$37$, because the sequence is $n+7$ and $-n^2/2+23n/2+16$ interleaved.</p>
<p>Use imagination and you come with equally good excuses for the other two options.</p>
|
2,816,365 | <p><a href="https://i.stack.imgur.com/T1W5r.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/T1W5r.jpg" alt="enter image description here"></a></p>
<p>I don't understand how it went from line four to five in the proof?
Do you need to use induction? We haven't covered it yet.</p>
| Asaf Karagila | 622 | <p>When you put a filter in your sink, the idea is that you filter out the big chunks of food, and let the water and the smaller chunks (which can—in principle—be washed through the pipes) go through.</p>
<p>You filter out the larger parts.</p>
<p>A filter filters out the larger sets. It is a way to say "these sets a... |
201,994 | <p>We arbitrarily choose n lattice points in a 3-dimensional Euclidean space such that no three points are on the same line. What is the least n in order to guarantee that there must be three points x, y, and z among the n points such that the center of mass of the x-y-z triangle is a lattice point?</p>
| joriki | 6,622 | <p>The answer $19$ has already been given, including references, so this is just a numerical check of that result.</p>
<p>The specific shape of the lattice is irrelevant, since a point is a lattice point iff it is an integer linear combination of the lattice vectors. The centre of mass of three lattice points is a lat... |
2,085,622 | <p>The question I am working through states:</p>
<p><em>Three digits are selected at random from the digits 1 through 10, without replacement. Find the probability that at least 2 digits are prime.</em></p>
<p>I feel as though I am close, but not quite hitting the mark with my solution:</p>
<p>My total number of sel... | Ross Millikan | 1,827 | <p>You are correct for the chance of selecting three prime numbers, but for selecting exactly two you need to multiply by the number of ways to select the non-prime.</p>
|
24,412 | <p>Trying to plot the following two functions to show points of intersection.</p>
<pre><code>2 x + y - 1 == 0,
x - y + 2 == 0
ContourPlot[{2 x + y - 1 == 0, x - y + 2 == 0}, {x, -5, 5}, {y, -5, 5}]
</code></pre>
<p>The above shows the plots, but I find it difficult to see the point of intersection. I suspect that th... | kglr | 125 | <pre><code>ContourPlot[{2 x + y - 1 == 0, x - y + 2 == 0}, {x, -5, 5}, {y, -5, 5},
MeshFunctions -> {Function[{x, y}, 2 x + y - 1 - (x - y + 2)]},
MeshStyle -> Directive[PointSize[Large], Red], Mesh -> {{0}}]
</code></pre>
<p><a href="https://i.stack.imgur.com/vmG8g.png" rel="nofollow noreferrer"><img sr... |
2,444,591 | <p>How do I prove that: $f(S ∩ T) \neq f(S) ∩ f(T)$ unless $f$ is one-to-one?</p>
<p>I tried finding contradicting example like considering the function $f(x)=x^2$ with $S = \{-2,-1,0,1,2\}$ and $T=\{0,1,2,3,4\}$ or using the function $f(x) = |x|$, but, it seems to be true. But, my book says that In general the afore... | nonuser | 463,553 | <p>Take any $y\in f(S \cap P)$, then exist $x \in S \cap P$ such that $y= f(x)$. So, since $x\in S$ we have $y\in f(S)$ and since $x \in P$ we have $y\in f(P)$ and thus $y\in f(S)\cap f(P)$. So $\boxed{f(S\cap P)\subseteq f(S)\cap f(P)}$. So here we don't need injective property.</p>
<p>Now suppose $f$ is injective.</... |
164,987 | <p>Find $(x, a, b, c)$ if $$x! = a! + b! + c!$$</p>
<p>I want to know if there are more solutions to this apart from $(x, a, b, c) = (3, 2, 2, 2)$.</p>
| Community | -1 | <p>First note that $a,b,c < x$, since $1 \leq n!$. This means that $a,b,c \leq x-1$. This implies that $$x! = a! + b! + c! \leq 3 (x-1)! \implies x \leq 3$$
Further, $x! = a! + b! + c! \geq 3 \implies x >2$. Hence, $x=3$ is the only possibility left.</p>
|
333,265 | <p>Can $|-x^2 | < 1 $ imply that $-1<x<1$?
My steps are as follows?
$$| -x^2| < 1 $$
$$-1<(-x^2)< 1 $$
$$-1<(-x^2)< 1 $$
$$-1<x^2< 1 $$
$$\sqrt{-1}<x< \sqrt 1 $$</p>
<p>I'm actually looking for the radius of convergence for the power series of $\frac{1}{1-x^2}$:</p>
<p>$$\fra... | wece | 65,630 | <p>$$|(-x)^2|<1$$
$$|x^2|<1$$
$$|x|^2<1$$
$$|x|<\sqrt{1}$$
$$|x|<1$$
$$-1<x<1$$
Seem more correct to me. (The $\sqrt{-1}$ isn't really well define).</p>
|
970,882 | <p>This is homework so no answers please</p>
<p>We have Multiplication map $F:O(n)\times O(n)\to O(n)$ defined as $F(A,B)=AB$ $F:O(n)\times O(n)\to O(n)$, where $O(n)=\{A\in M(n\times n):AA^{t}=id\}$.
The smooth structure of O(n) comes from ImpFT: consider smooth map $f:M(n\times n)\to Sym(n\times n)$ defined as $f(A)... | Jack Lee | 1,421 | <p>Two basic theorems in differential geometry would be useful here. Suppose $M,N$ are smooth manifolds and $f\colon M\to N$ is a smooth map.</p>
<ol>
<li><strong>Restricting the domain:</strong> If $S\subseteq M$ is an (immersed or embedded) smooth submanifold, then $f|_M\colon S\to N$ is smooth.</li>
<li><strong>Res... |
2,961,592 | <p>We know 4 is a real number but how can we prove that it is a complex number? How can be describe it in the a+ib form??</p>
| Peter Szilas | 408,605 | <p><span class="math-container">$0<x<y$</span>.</p>
<p><span class="math-container">$(y-x)=$</span></p>
<p><span class="math-container">$(√y-√x)(√y+√x)>0.$</span></p>
<p>Since <span class="math-container">$√y+√x >0$</span>, we get </p>
<p><span class="math-container">$√y -√x>0$</span>, or <span clas... |
2,961,592 | <p>We know 4 is a real number but how can we prove that it is a complex number? How can be describe it in the a+ib form??</p>
| lhf | 589 | <p><span class="math-container">$
\sqrt{x} <\sqrt{y}
\implies
x = \sqrt{x} \sqrt{x} < \sqrt{x} \sqrt{y} = \sqrt{xy}
$</span></p>
<p><span class="math-container">$
\sqrt{x} <\sqrt{y}
\implies
\sqrt{xy} = \sqrt{x} \sqrt{y} < \sqrt{y} \sqrt{y} = y
$</span></p>
|
1,379,899 | <p>I'm trying to achieve a better conception of what it means to "divide out" a variable/number, because I'm currently have a lot of trouble justifying to myself why it actually works the way it does in certain contexts. I apologize if this is too much of an elementary question but I can't seem to come to a satisfying ... | Michael Hardy | 11,667 | <p>Look at the number of permutations of $3$ things out of $5$. Call the five things $ABCDE$.
$$
\begin{array}{c|c|ccc}
\overbrace{\begin{array}{cccc} ABC & ACB & ADB & AEB \\ ABD & ACD & ADC & AEC \\ \underbrace{ABE}_B & \underbrace{ACE}_C & \underbrace{ADE}_D & \underbrace{AED}_E ... |
569,814 | <p>I was watching a video on YouTube on Quantum Mechanics Concepts and saw that if you wanted to convert a probability amplitude to a probability, you square it. In the video he said that this was equivalent to multiply by it's complex conjugate. So is this correct? Is squaring the same as multiplying by the complex co... | Community | -1 | <p>No, it is not: Given $i$, we have</p>
<p>$$i^2 = -1 \ne 1 = i \overline{i}$$</p>
<p>What we can say, however, is that the absolute value of any complex $z$ is defined to be</p>
<p>$$|z| = \sqrt{z \overline{z}}$$</p>
|
1,850,978 | <p>My question concerns the following problem:</p>
<blockquote>
<p>Let $K=\mathbb F_7[T]/(T^3-2)$. Show that $X^3-2$ splits into linear factors in $K[X]$.</p>
</blockquote>
<p>Write $K\simeq \mathbb F_7[\alpha]$ for a root $\alpha\in \overline{\mathbb F_7}$ of $X^3-2$. I found out through experimentation that $2\al... | lhf | 589 | <p>$X^3-2$ is irreducible mod $7$. This implies that $K$ has $7^3$ elements.</p>
<p>Since the group $K^\times$ is cyclic of order $7^3-1= 2 \cdot 3^2 \cdot 19$, there is an element $\omega \in K^\times$ of order $3$.</p>
<p>Then $X^3-2=(X-\alpha)(X-\omega\alpha)(X-\omega^2\alpha)$.</p>
<p>As you have discovered, you... |
1,900,365 | <p>Apparently this is a true statement, but I cannot figure out how to prove this. I have tried setting </p>
<p>$$(m)(m + 1)(m + 2)(m + 3) = (m + 4)^2 - 1 $$</p>
<p>but to no avail. Could someone point me in the right direction? </p>
| Steven Alexis Gregory | 75,410 | <p>I will show that
$16m(m+1)(m+2)(m+3) = 16(m^2 + 3m + 1)^2 - 16$</p>
<p>It will follow that
$$m(m+1)(m+2)(m+3) = (m^2 + 3m + 1)^2 - 1$$</p>
<p>Proof:
\begin{align}
16 m(m+1)(m+2)(m+3)
&=(2m)(2m+2)(2m+4)(2m+6)\\
& ---\text{let $2m = n-3$}---\\
&=(n-3)(n-1)(n+1)(n+3)\\
&=(n-3)(n+3) \cdo... |
171,602 | <p>In all calculus textbooks, after the part about successive derivatives, the $C^k$ class of functions is defined.
The definition says : </p>
<blockquote>
<p>A function is of class $C^k$ if it is differentiable $k$ times and the $k$-th derivative is continuous.</p>
</blockquote>
<p>Wouldn't be more natural to defi... | Community | -1 | <p>One reason is that it's desirable to equip a class of functions with a <a href="http://en.wikipedia.org/wiki/Norm_%28mathematics%29">norm</a>, particularly with a norm which makes the class a complete metric space, i.e., <a href="http://en.wikipedia.org/wiki/Banach_space">Banach space</a>. In $C^k$ we can use the su... |
1,650,638 | <p>I have a sequence of reals $S = s_1,s_2,\dots,s_n$ such that $s_i-s_{i-1}$ is a Gaussian distribution. From histogram of sequence $S$ (10000 elements) it appears that it is uniform distribution. Is it true ? If yes, can we prove it ? If no, what can we say about $s_i$ ?</p>
| Martin Argerami | 22,857 | <p>A linear operator is not hermitian "with respect to a basis". </p>
<p>When there is a basis of eigenvectors as you mention, $Av_k=\lambda_kv_k$, then $A$ is already diagonal. The point of diagonalization is to find such basis. </p>
|
4,114,962 | <blockquote>
<p>Prove that the product of four consecutive natural numbers are not the square of an integer</p>
</blockquote>
<p>Would appreciate any thoughts and feedback on my suggested proof, which is as follows:</p>
<p>Let <span class="math-container">$f(n) = n(n+1)(n+2)(n+3) $</span>.
Multiplying out the expressio... | jjagmath | 571,433 | <p>Hint: <span class="math-container">$n(n+1)(n+2)(n+3) +1$</span> is a square.</p>
|
2,539,327 | <p>The equation of a parabola is given by:
$y= ax^2 + bx+c$</p>
<p>Why is it that when the coefficient of $x$ i.e. $a$ is positive we get an upward parabola and when it's negative we get a downward parabola? </p>
<p>Also, I <a href="https://www.desmos.com/calculator/ldyxmzfrws" rel="nofollow noreferrer">saw</a> that... | Dr. Sonnhard Graubner | 175,066 | <p>let $$f(x)=ax^2+bx+c$$ then we have $$f'(x)=2ax+b$$ and $$f''(x)=2a$$ if $$a>0$$ then we get a Minimum Point, if $a<0$ then we get a Maximum Point.</p>
|
2,539,327 | <p>The equation of a parabola is given by:
$y= ax^2 + bx+c$</p>
<p>Why is it that when the coefficient of $x$ i.e. $a$ is positive we get an upward parabola and when it's negative we get a downward parabola? </p>
<p>Also, I <a href="https://www.desmos.com/calculator/ldyxmzfrws" rel="nofollow noreferrer">saw</a> that... | Glorious Nathalie | 948,761 | <p>You're given <span class="math-container">$y = a x^2 + b x + c $</span></p>
<p>Start your analysis by completing the square in <span class="math-container">$x$</span>, as follows:</p>
<p><span class="math-container">$\begin{equation}
\begin{split}
y &= a ( x^2 + \dfrac{b}{a} x ) + c \\
&= a (x + \dfrac{b}{... |
3,490,454 | <p>Definition from Munkres, the textbook we are following: If <span class="math-container">$Y$</span> is a compact Hausdorff space and <span class="math-container">$X$</span> is a proper subspace of <span class="math-container">$Y$</span> whose closure equals <span class="math-container">$Y$</span>, then <span class="m... | AHusain | 277,089 | <p>Think about just one interval <span class="math-container">$(0,2\pi)$</span> first. If you add 2 points, you could make the compact <span class="math-container">$[0,2\pi]$</span>. But the one point compactification does it differently. It only adds one point so that the space becomes a circle like <span class="math-... |
4,338,558 | <p>I am not sure if my proof here is sound, please could I have some opinions on it? If you disagree, I would appreciate some advice on how to fix my proof. Thanks</p>
<p><span class="math-container">$X_1, X_2, ..., X_n$</span> are countably infinite sets.</p>
<p>Let <span class="math-container">$X_1 = \{{x_1}_1, {x_1}... | H A Helfgott | 169,068 | <p>Let me treat the case of <span class="math-container">$F(x,y) = x^a y^a \max(x,y)$</span>, <span class="math-container">$a>1$</span>. (In the above, <span class="math-container">$a=5/3$</span>.)</p>
<p>We let <span class="math-container">$u = x y$</span>. Then our double integral equals 2 I, where
<span class="ma... |
110,867 | <p>I want to prove that the function $f(x) := x^3$ for all real $x$ defines a homeomorphism from $\mathbb{R}$ to $\mathbb{R}$. But I am finding it difficult to prove that the inverse map is continuous!</p>
| Henrik | 22,705 | <p>HINT: As far as I know you could use the very powerful tool that any continuous bijection on a compact topological space onto a topological Hausdorff space is a homeomorphism. </p>
|
4,138,564 | <p>Prove that if <span class="math-container">$q_i$</span>'s are polynomials of degree <span class="math-container">$i$</span>,then <span class="math-container">$q_0,...,q_n$</span> are linearly independent.</p>
<hr />
<p>Assume they are not linearly independent, so there is a scalar <span class="math-container">$\lamb... | Community | -1 | <p>As the matrix of coefficients is triangular with a nonzero diagonal, you can reduce to canonical polynomials by Gaussian elimination. And the canonical basis <span class="math-container">$is$</span> of course made of independent vectors. (Note that as <span class="math-container">$q_0$</span> is not included, the <s... |
3,143,532 | <blockquote>
<p>Find <span class="math-container">$\det A$</span> and <span class="math-container">$\text{Tr} A$</span> for the matrix <span class="math-container">$A\in M_n(\mathbb{Q})$</span> such that <span class="math-container">$\sqrt[n]{p}$</span> is an eigenvalue of <span class="math-container">$A$</span>, whe... | J.G. | 56,861 | <p>To complement Zacky's answer I'll add a proof that <span class="math-container">$I(0)=0$</span>. Applying <span class="math-container">$x\mapsto\frac{1}{x}$</span> for <span class="math-container">$x\ge 1$</span> gives <span class="math-container">$$I(x)=2\int_0^\infty\ln|1-x^{-2}|dx=2\int_0^1\left[(1+\frac{1}{x^2})... |
1,552,775 | <p>In how many ways 5 blue pens and 6 black pens can be distributed to 6 children?</p>
<hr>
<p>To do that I used:
$\text{Coefficient of } x^6 \text{ in } ((1+x+x^2+x^3+x^4+x^5)(1+x+x^2+x^3+x^4+x^5+x^6))$
and got answer = 6</p>
<p>but options given are:</p>
<p>a) 97020</p>
<p>b) 116424</p>
<p>c) 8008</p>
<p>d) 67... | seed | 294,587 | <p>If you denote by $B^n_m$ the number of ways to distribute n pens among m children, you can write down its generating series:
$$
\sum_{n=0}^\infty{B_m^nx^n}=(1+x+x^2+\ldots)^m=(1-x)^{-m}
$$
Now by taking the coefficient at $x^n$ you find a formula for $B_m^n$:
$$
B_m^n=(-1)^n C_{-m}^n=C_{m+n-1}^n
$$
The answer to th... |
3,676,719 | <p>I've recently started to learn about differential equations and I am having a hard time solving any of them.</p>
<p>I feel like I'm missing some steps.</p>
<p>Therefore, could I please know how to solve:</p>
<p><span class="math-container">$(ye^x +y)dx+ye^{(x+y)}dy=0$</span></p>
<p>So far, I've looked around the... | Ben Grossmann | 81,360 | <p>In general, the statement "if <span class="math-container">$A^2 = 0$</span>, then <span class="math-container">$\operatorname{tr}(A) = 0$</span>" can be proven using the fact that the trace is the sum of the eigenvalues. One proof for the specific case of <span class="math-container">$2 \times 2$</span> matrices is ... |
285,975 | <p>I'm not sure I understand what to do with what's given to me to solve this. I know it has to do with the relationship between velocity, acceleration and time.</p>
<blockquote>
<p>At a distance of <span class="math-container">$45m$</span> from a traffic light, a car traveling <span class="math-container">$15 m/sec$</... | Ishan Banerjee | 52,488 | <p>$f(kc)$ where $k\in \mathbb{N}$ and $c$ is the constant term of the polynomial is nonprime. Not sure what to do if $c=1$ or $c=-1$ though.</p>
|
285,975 | <p>I'm not sure I understand what to do with what's given to me to solve this. I know it has to do with the relationship between velocity, acceleration and time.</p>
<blockquote>
<p>At a distance of <span class="math-container">$45m$</span> from a traffic light, a car traveling <span class="math-container">$15 m/sec$</... | Math Gems | 75,092 | <p><strong>Hint</strong> $\ $ For some $\,x_0\,$ we have $\,f(x_0)= m\ne \pm1\: $ (see the note below for one proof). </p>
<p>It follows that $\,m\mid f(mn+x_0)\,$ since $\,{\rm mod\ } m\!:\ f(mn+x_0)\equiv f(x_0)\equiv m \equiv 0.\:$ </p>
<p><strong>Note</strong> $\ $ If $\,f(x) = \pm 1\,$ for all $\,x\in \Bbb Z\,$... |
2,102,619 | <p><strong>What is the total running time of counting from 1 to n in binary if the time needed to add 1 to the current number i is proportional to the number of bits in the binary expansion of i that must change in going from i to i + 1?</strong></p>
<p>Having trouble getting my head a around this one because I cannot... | Narasimham | 95,860 | <p>It is advantageous to recognize slope $\tan\phi$ as parameter.
$$ f^{\prime}== \tan \phi \tag1$$
$$ f^{\prime \prime}== \sec^2 \phi \frac{d\phi}{dx} \tag2 $$
$$ \sec^3 {\phi} = \sec^2 \phi \frac{d\phi}{dx} \tag3 $$
$$ \cos \phi\, d\phi = dx \tag4 $$
$$\sin\phi = x+c = x, \tag5 $$
with given BC. Now letting $y... |
3,987,911 | <p>In the game of bridge, a player is given 13 cards from a deck of 52 cards, what is the probability that he/she gets exactly one king and one queen? Furthermore what is the expected value of the number of aces he/she gets?</p>
| ETS1331 | 418,667 | <p>We rephrase the problem: the bug does not teleport, but instead heads from <span class="math-container">$(0,-63)$</span> to <span class="math-container">$(x,0)$</span> at <span class="math-container">$\sqrt{2}$</span> units per second and then heads from <span class="math-container">$(x,0)$</span> to <span class="ma... |
1,644,130 | <p>I proved for a bounded set $\Omega$ and $1 \leq p \leq q \leq \infty$ that $L^{q}(\Omega) \subset L^{p}(\Omega)$. What is an example that would show strict inclusion, $ p<q$, and false if $\Omega$ is not bounded?</p>
| ZFR | 99,694 | <p>Counterexample: $f(x)=\frac{1}{x}\in L^2([1,+\infty))$, but it does not belong to $L^1([1,+\infty))$</p>
|
2,755,091 | <p>I know that I have to prove this with the induction formula. If proved the first condition i.e. $n=6$ which is divisible by $6$. But I got stuck on how to proceed with the second condition i.e $k$.</p>
| Dr. Sonnhard Graubner | 175,066 | <p>We have $$7\equiv 1\mod 6$$ then $$7^n\equiv 1^n=1\equiv 1\mod 6$$ so $$7^n-1\equiv 0 \mod 6$$</p>
|
15,994 | <p>I am trying to solve this problem <a href="https://mathematica.stackexchange.com/questions/15922/help-me-to-solve-a-condition-from-an-equation-that-requires-many-reductions">here</a> but have to understand basics. Suppose I have a multiplication such as <span class="math-container">$a*(b+c+d+...+x)$</span>. How can ... | cartonn | 4,593 | <p>I know this has been answered in the comments, but just for the sake of having an actual answer..</p>
<pre><code>Expand[a(b+c)]
</code></pre>
<p>or </p>
<pre><code>Expand[(a+b)(b+c)]
</code></pre>
|
2,408,141 | <p>Let $X\to Y$ so that $f(x)=x^2+4$ </p>
<p>$X=\{6,9,2,8,5\}$ and $Y=\{27,85,40,8,12,29,63,68,17\}$</p>
<p>a) state the domain of $f$</p>
<p>b) state the range of $f$</p>
<p>I have calculated that the domain of $f(x)$ is all real numbers and that the range of $f(x)$ is $f\ge4$.</p>
<p>Therefore I thought that the... | Eric Towers | 123,905 | <p>Let $A,B$ be an algebraic objects of type $T$. Let the algebraic operations $o_1, \dots, o_k$ for some positive integer $k$ be the operations of $T$. Let the $T$-operations in $A$ be denoted $o_j^A$ and the $T$-operations in $B$ be denoted $o_j^B$ for $j = 1 \dots k$. Suppose $A$ is generated as a $T$ by $a_1, \d... |
3,621,836 | <p>Let's say I have a large determinant with scalar elements:</p>
<p><span class="math-container">$$\begin{vmatrix} x\cdot a & x\cdot b & x\cdot c & x\cdot d \\ x\cdot e & x\cdot f & x\cdot g & x\cdot h \\ x\cdot i & x\cdot g & x\cdot k & x\cdot l \\ x\cdot m & x\cdot n & x\... | Alan | 696,271 | <p>Clearly we have that <span class="math-container">$1, \omega, \omega^2$</span> are the roots of the polynomial <span class="math-container">$p(x):=x^3-1$</span>.
Therefore, it follows that
<span class="math-container">$$p(x)=(x-1)(x-\omega)(x-\omega^2)$$</span>
and that <span class="math-container">$\omega^3=1$</sp... |
244,521 | <p>I cannot prove this statement, I tried to prove by using the definition of open sets however i feel that it is necessary prove it in two directions since it's an iff statement.</p>
<p>The question is, </p>
<p>Let $X$ be a metric space with metric $d$, and let $x_n$ be a sequence of points in $X$. Prove that $x_n\r... | robjohn | 13,854 | <p><strong>Correction</strong></p>
<p>The condition $B(a,r)\cap U\ne\varnothing$ is used to show that $a$ is a limit point of $U$. To show that $U$ is open,
$$
\forall a\in U,\exists r>0:B(a,r)\subset U
$$</p>
<p><strong>Hint</strong></p>
<p>The standard metric space concept of convergence is
$$
\forall\epsilon&g... |
3,774,263 | <ol>
<li>Is it possible that distance (<span class="math-container">$r$</span>) or angle (<span class="math-container">$θ$</span>) contains Imaginary or Complex number?</li>
<li>If the answer is yes, how can I convert a number like that (Polar with complex argument) to Rectangular number? <br />
For example:
<strong><s... | Kavi Rama Murthy | 142,385 | <p>Any non-zero continuous linear functional is surjective. If the kernel is <span class="math-container">$\{0\}$</span> then it is an isomorphism onto <span class="math-container">$\mathbb R$</span>. Hence <span class="math-container">$X$</span> is one-dimensional.</p>
<p>Proof of surjectivity: Choose <span class="m... |
2,082,062 | <p>In Avner Friendman's Mondern Analysis book, he makes a statement that has stumped me when proving that for $\{f_n\}$ a sequence of measurable functions, $\sup_n f_n$ And $\inf f_n$ Are measurable.</p>
<blockquote>
<p>The assertion for $\inf_n f_n $ follows from $\inf (f_n )=- \sup (-f_n)$ .</p>
</blockquote>
<p>... | Patrick Abraham | 337,503 | <p>May $f$ be a linear functional.
Then $f$ due it's linearity fulfils the homogeneity and additivity properties.</p>
<p>Let $a \in \mathbb R$ and let $v$ be a vector:
$$f(a\vec v) = a f(\vec v)$$</p>
<p>Hint:</p>
<p>Is there a different approach to write $-f$ with a scalar $a$ as $a f$.</p>
|
1,136,109 | <p>Suppose that $a,b$ are reals such that the roots of $ax^3-x^2+bx-1=0$ are all positive real numbers. Prove that: </p>
<p>$(i)~~0\le 3ab\le 1$<br>
$(ii)~~b\ge \sqrt3$. </p>
<p>My attempt: </p>
<p>I could solve the first part by Vieta's theorem. But, I am stuck on the second part. Please help. Thank you.</p>
| DeepSea | 101,504 | <p><strong>Hint</strong>: $x_1^3 +x_2^3 + x_3^3 = (x_1+x_2+x_3)^3 - 3(x_1+x_2)(x_2+x_3)(x_3+x_1) \leq (x_1+x_2+x_3)^3 - 24x_1x_2x_3$, by AM-GM inequality, and write:</p>
<p>$b = \dfrac{3+\displaystyle \sum x_i^2-a\displaystyle \sum x_i^3}{\displaystyle \sum x_i}$. Express $b$ as a function of $a$ and find the critical... |
1,963,214 | <p>$\displaystyle\sum_{i=0}^{k-1}2^i=2^k-1$ for all $k \in\Bbb N$.</p>
<p>Clearly, the first step here is easy. You can start with k=1 and solve to get $2^0=2^1-1$.</p>
<p>What is a bit more challenging is the induction step. I don't even know where to begin here. Where would I even begin?</p>
| Community | -1 | <blockquote>
<p>How do (I) begin proving... </p>
</blockquote>
<p>There is always something you could try. For instance, try to <a href="https://en.wikipedia.org/wiki/How_to_Solve_It" rel="nofollow">solve some simpler related questions</a>:</p>
<ul>
<li>Can you write down by <strong><em>definition</em></strong> wha... |
1,145,902 | <p>I've searched for the inner product definition and I saw that it should only satisfy some conditions (axioms) and thus there could be several operations representing an inner product, one of which is the usual multiplication, or the dot product.
the question is: why is the inner product of 2 functions is defined by ... | YTS | 126,222 | <p>Yes you are right.
And $||x||$ is just $|x|$ since the absolute value is always non- negative. In general you have that $|g(x)|=g(x)$ if and only if $g(x)$ is non-negative. </p>
|
2,536,206 | <p>Let $M$ be a smooth manifold of dimension $n$, and let $T(M)$ be the tangent bundle of $M$. Let $F$ be a smooth vector field, which is to say a smooth section of the canonical map $T(M) \rightarrow M$. If I understand the manifold structure on $T(M)$ correctly, smoothness means that for any chart $(U,\phi)$ of $M$... | rtybase | 22,583 | <p>From the set theory perspective, let's note
$$P \overset{\text{def}}{=}\{p \in \mathbb{N} \mid p \text{ - prime}\}$$
$$P_{\leq n} \overset{\text{def}}{=} \{p \in P \mid p \leq n \}$$
$$M_n \overset{\text{def}}{=} \{p \in P \mid p \mid n\}$$
$\color{red}{M_n \ne \varnothing, \forall n\geq2}$. Then, it's easy to show... |
3,624,732 | <p>I'm studying logical operators for school and there's a weird question that keeps bugging me even though it seems pretty basic.
<br>
I was asked to evaluate the proposition : <strong>p -> q -> r</strong> with p, r are False and q is True.
<br>
I tried evaluating it from left to right like this: <strong>( ( p -> q ) ... | user400188 | 400,188 | <p>The full convention is as follows: Outermost parentheses may be omitted, and in the absence of parentheses, the order of operations is</p>
<ol>
<li><span class="math-container">$\lnot$</span></li>
<li><span class="math-container">$\land$</span></li>
<li><span class="math-container">$\lor$</span></li>
<li><span clas... |
2,365,797 | <p>For integers $n\geq 1$ let $\mu(n)$ the Möbius function, and let $\psi(z)$ the Digamma function, see its definition and how is denoted and typeset in Wolfram Language, if you need it from this <a href="http://mathworld.wolfram.com/DigammaFunction.html" rel="nofollow noreferrer">MathWorld</a>.</p>
<p>While I was doi... | Claude Leibovici | 82,404 | <p><em>This is not an answer but it is too long for a comment.</em></p>
<p>To confirm the result, I computed
$$S_k=\sum_{n=1}^{10^k}\frac{\mu(n)}{n}\psi\left(1+\frac{1}{n}\right)$$ and obtained the following decimal representations
$$\left(
\begin{array}{cc}
k & S_k \\
1 & 0.4606256866 \\
2 & 0.48255387... |
2,365,797 | <p>For integers $n\geq 1$ let $\mu(n)$ the Möbius function, and let $\psi(z)$ the Digamma function, see its definition and how is denoted and typeset in Wolfram Language, if you need it from this <a href="http://mathworld.wolfram.com/DigammaFunction.html" rel="nofollow noreferrer">MathWorld</a>.</p>
<p>While I was doi... | Community | -1 | <p>Since $\sum_{n=1}^\infty\frac{\mu(n)}{n}=0$, we have $$\sum_{n=1}^\infty\frac{\mu(n)}{n}\psi\left(1+\frac{x}{n}\right)=\sum_{n=1}^\infty\frac{\mu(n)}{n}\left[\psi\left(1+\frac{x}{n}\right)+\gamma\right],$$ and the series is uniformly convergent on $[0,1]$. Now
$$\sum_{n=1}^\infty\frac{\mu(n)}{n}\left[\psi\left(1+\fr... |
4,168,198 | <p>If M is a connected n-manifold and N is a codim 2 submanifold
then M-N is connected.</p>
<p>Is this true?</p>
<p>I want to show <span class="math-container">$H_0(M-N)=\mathbb Z$</span>. I think it's better to do in <span class="math-container">$\mathbb Z_2$</span> coefficient because we get orientability. But becaus... | user939711 | 939,711 | <p>The goal of this answer is to use Moishe Kohan's idea to prove the general statement for not-necessarily-closed submanifolds. In fact even more than that can be said.</p>
<p>Let <span class="math-container">$M$</span> be a metrizable topological manifold and let <span class="math-container">$f: S \to M$</span> be a ... |
478,912 | <p>Ok so I get the basics of this, I just can't put it all together. $U$ is contained in the cartesian product of $\mathbb R^n \times \mathbb R$. What is $\mathbb R^n \times \mathbb R$ though? I know what a cartesian product is, but this seems weird to me. Someone please help explain this part, thanks!</p>
| Shuchang | 91,982 | <p>You could understand it as space-time which $R^n$ is space dimension and the other $R$ is for time. In this view, $U=\{(x,t)\}$ contains points representing an event occuring at point $x$ and time $t$. Does it help?</p>
|
2,936,356 | <p>Let <span class="math-container">$a,b>0$</span>. How can I calculate the roots of the following polynomial?</p>
<p><span class="math-container">$$2bx^6 + 4bx^5+(4b-a)x^4 + 4(b+a)x^3 + (2b-6a)x^2+4ax-a=0$$</span></p>
| Robert Z | 299,698 | <p>Note that the given equation is equivalent to
<span class="math-container">$$s(x-1)^4=2x^2(x+1)^2(x^2+1).$$</span>
where <span class="math-container">$s=a/b>0$</span>.
I guess there is no "easy" closed formula for the roots, but, for example, we always have at least two real roots: one in <span class="math-contai... |
28,348 | <p>Thomson et al. provide a proof that <span class="math-container">$\lim_{n\rightarrow \infty} \sqrt[n]{n}=1$</span> in <a href="http://classicalrealanalysis.info/documents/TBB-AllChapters-Landscape.pdf#page=95" rel="noreferrer">this book (page 73)</a>. It has to do with using an inequality that relies on the binomial... | Paramanand Singh | 72,031 | <p>Let $n > 1$ so that $n^{1/n} > 1$ and we put $n^{1/n} = 1 + h$ so that $h > 0$ depends on $n$ (but we don't write the dependence explicitly like $h_{n}$ to simplify typing) Our job is done if show that $h \to 0$ as $n \to \infty$.</p>
<p>We have $$n = (1 + h)^{n} = 1 + nh + \frac{n(n - 1)}{2}h^{2} + \cdots... |
28,348 | <p>Thomson et al. provide a proof that <span class="math-container">$\lim_{n\rightarrow \infty} \sqrt[n]{n}=1$</span> in <a href="http://classicalrealanalysis.info/documents/TBB-AllChapters-Landscape.pdf#page=95" rel="noreferrer">this book (page 73)</a>. It has to do with using an inequality that relies on the binomial... | pre-kidney | 34,662 | <p>The limit follows from these inequalities and the squeeze theorem:
<span class="math-container">$$
1<n^{1/n}<1+\sqrt{\frac{2}{n-1}},\qquad n>1
$$</span>
where the right inequality follows by keeping only the third term in the binomial expansion:
<span class="math-container">$$
(1+x)^n>\binom{n}{2}x^2= n,... |
4,434,225 | <p>Prove or disprove, "if <span class="math-container">$f$</span> is concave function, then <span class="math-container">$|f|$</span> is also concave".</p>
<p>I know the result is false for conves function, but for concave function, I guess it is true. But I am unable to do the proof.</p>
<p>I have tried to s... | Dr. Sundar | 1,040,807 | <p>Counterexample:</p>
<p>Note that
<span class="math-container">$$
f(x) = - x^2, \ \ \ (x \in \mathbf{R})
$$</span>
is a concave function, but
<span class="math-container">$$
| f(x) | = x^2, \ \ \ (x \in \mathbf{R})
$$</span>
is a convex function.</p>
|
2,010,423 | <p>Are $$(a, b, c) = (\mid 1 \mid, \mid 2 \mid, \mid 2 \mid), (\mid 2 \mid, \mid 4 \mid, \mid 4 \mid), (\mid 2 \mid, \mid 3\mid, \mid 6 \mid), (\mid 1 \mid, \mid 1 \mid, \mid 1 \mid)$$ the only integers such that
$$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} $$ is an integer ?</p>
| Reese Johnston | 351,805 | <p>Assuming we only care about positive integers for the time being, notice that if $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$ is an integer then it must be at least $1$. If three numbers add up to at least $1$, then at least one of them must be at least $\frac{1}{3}$ - so at least one of $a$, $b$, and $c$ must be no mo... |
324,307 | <p><a href="http://en.wikipedia.org/wiki/Tucker%27s_lemma" rel="nofollow">Tucker's Lemma is here.</a></p>
<p>Let's stay within the 2D case for now. A standard proof is constructive:</p>
<p>(1) Pick an arced edge on the boundary of the circle. Note its labeling (for example, (1, 2)).</p>
<p>(2) Walk into the circle... | Thomas Andrews | 7,933 | <p>As commenters have noted, you can't conclude $a=c=1$.</p>
<p>The condition $ad+b=bc+d$ can be rewritten as:</p>
<p>$$d(a-1)=b(c-1)$$</p>
<p>Now, if $a=1$ then $b\neq 0$ (or else $f(x)=x$ is the identity.)</p>
<p>So in that case, then $x+b$ commutes with $cx+d$ if and only if $c=1$. So if $x+b$ commutes with $cx+... |
4,214,631 | <p>I'm trying to understand a paragraph from the article <a href="https://www.sciencedirect.com/science/article/pii/S0022123600936875" rel="nofollow noreferrer">Christ, Kiselev: Maximal functions associated to filtrations</a>. After the proof of Theorem 1.1., the authors deduce a theorem of Menshov as a corollary:</p>
... | Jose27 | 4,829 | <p>This is a standard argument in harmonic analysis. Let me give the abstract version first.</p>
<p>Suppose we have a sequence of linear operators <span class="math-container">$T_n$</span> mapping measurable functions on <span class="math-container">$(X, \mu)$</span> to measurable functions on <span class="math-contain... |
405,830 | <p>Are there any cases of finite subgroups of <span class="math-container">$O_n(\mathbb{Q})$</span> <s>not contained in</s> <i>not isomorphic to any subgroup of</i> <span class="math-container">$O_n(\mathbb{Z})$</span>?</p>
| Noam D. Elkies | 14,830 | <p>If it's the same <span class="math-container">$n$</span> then yes this can happen. For example, the lattice <span class="math-container">$D_4$</span> (consisting of all integer vectors in <span class="math-container">${\bf Z}^4$</span> with even sum) has more isometries than <span class="math-container">${\bf Z}^4$... |
54,088 | <p>I am studying KS (Kolmogorov-Sinai) entropy of order <em>q</em> and it can be defined as</p>
<p>$$
h_q = \sup_P \left(\lim_{m\to\infty}\left(\frac 1 m H_q(m,ε)\right)\right)
$$</p>
<p>Why is it defined as supremum over all possible partitions <em>P</em> and not maximum? </p>
<p>When do people use supremum and w... | Community | -1 | <p>Consider for example the set $X = (0,1)$. Then $\sup X=1$ but $\max X$ does not exist. </p>
<p>Generally, for a set $X\subset {\mathbb R}$, we say $x=\max X$ if $\color{blue}{x\in X}$ and
$$\forall y\in X, y\leq x.$$</p>
<p>We have $x=\sup X$ if
$$\forall y\in X, y\leq x$$
and $$\forall\epsilon>0,\quad\exists ... |
4,517,063 | <p>Prove that there are exactly <span class="math-container">$8100$</span> different ways of distributing <span class="math-container">$4$</span> indistinguishable black marbles and <span class="math-container">$6$</span> distinguishable coloured marbles ( none of them black) into <span class="math-container">$5$</span... | true blue anil | 22,388 | <p>Just distribute the distinguishable balls as a product of two multinomial coefficients,<br />
<em>[laying down a pattern</em>]<span class="math-container">$\;\times\;$</span> [<em>permuting it</em>]<br />
and <strong>forget about filler indistinguishable black balls</strong>, thus</p>
<p><span class="math-container"... |
2,417,029 | <p>I have been told that a line segment is a set of points.
How can even infinitely many point, each of length zero, can make a line of positive length?</p>
<p>Edit:
As an undergraduate I assumed it was due to having uncountably many points.
But the Cantor set has uncountably many elements and it has measure $0$.</p>
... | Caleb Stanford | 68,107 | <p>I agree with Mikhail Katz: you were right as an undergraduate when you "assumed it was due to having uncountably many points." Since you say you have learned measure theory, I am not sure why you discarded this (correct) explanation.</p>
<p>To those who are less familiar with differing infinite cardinalities, a sim... |
657,992 | <p>I understand how to explain but can't put it down on paper.</p>
<p>$\displaystyle \frac{a}{b}$ and $\displaystyle \frac{c}{d}$ are rational numbers. For there to be any rational between two numbers I assume $\displaystyle \frac{a}{b} < \frac{c}{d}$.</p>
<p>I let $\displaystyle x = \frac{a}{b}$ and $\displaystyl... | vonbrand | 43,946 | <p>For rationals $r_1$ and $r_2$ you can always find the rational $(r_1 + r_2) / 2$ between them. As you can repeat this at will, you can construct infinitely many rationals between them (essentially the set $r_1 + (r_2 - r_1) \cdot 2^{-n - 1}$). </p>
|
394,489 | <p>$$\int^L_{-L} x \sin(\frac{\pi nx}{L})$$</p>
<p>I've seen something like this in Fourier theory, but I'm still not sure how to approach this integral. Wolfram Alpha gives me the answer, but no method. Integrate by parts? Substitution?</p>
| apnorton | 23,353 | <p>Whenever you have $p(x)f(x)$, where $p(x)$ is a polynomial, and $f(x)$ is a function like $\sin(x)$ or $e^x$, such that differentiating doesn't increase its "order," integration by parts is always a good technique. Differentiate the polynomial so it becomes "less complicated," and integrate the other function.</p>
... |
162,364 | <p>Assuming we have a list below, it has real number elements and complex numbers, how can I quickly find if the list has any Real number that its value is less than 1.0? </p>
<pre><code> lis = {3 Sqrt[354], Sqrt[2962], Sqrt[2746], 3 Sqrt[282], Sqrt[2338],
Sqrt[2146], 3 Sqrt[218], Sqrt[1786], Sqrt[1618], 27 Sqrt[2]... | Nasser | 70 | <blockquote>
<p>how can I quickly find if the list has any Real number that its value
is less than 1.0?</p>
</blockquote>
<p>One way might be</p>
<pre><code>Cases[ReIm[lis],{x_,y_}/;y==0&&x<1]
(*{} *)
</code></pre>
<p>Example</p>
<pre><code> lis={.7, .7I .7+1, 2, 2I, 2+1 I};
Cases[ReIm[lis],{x_,y_}/... |
162,364 | <p>Assuming we have a list below, it has real number elements and complex numbers, how can I quickly find if the list has any Real number that its value is less than 1.0? </p>
<pre><code> lis = {3 Sqrt[354], Sqrt[2962], Sqrt[2746], 3 Sqrt[282], Sqrt[2338],
Sqrt[2146], 3 Sqrt[218], Sqrt[1786], Sqrt[1618], 27 Sqrt[2]... | Michael E2 | 4,999 | <p>Here's a way using <code>Pick</code>:</p>
<pre><code>Pick[lis, UnitStep[1 - Re@#] + I*Im@# &@N@lis, 1]
(* {} *)
lis2 = lis/10;
Pick[lis2, UnitStep[1 - Re@#] + I*Im@# &@N@lis2, 1]
(* {Sqrt[29/2]/5, 3/(5 Sqrt[2]), 0.109545, Sqrt[29/2]/5, 0.102956} *)
</code></pre>
<p><code>UnitStep[1 - Re@#]</code> retu... |
1,387,216 | <p>Let $X$ be a metric space, and let $E$ and $K$ be two sets such that $E\subset K$.</p>
<p>I want to prove:</p>
<p>If $p$ is a limit point of $E$, then $p$ is a limit point of $K$.</p>
<p>Proof: if every neighborhood of $p\in E$ contains a point $q\neq p$ such that $q\in E$, then this $q$ is also in $K$, then $p$ ... | coldnumber | 251,386 | <p>You have the right idea, but I think a piece is missing.</p>
<p>Yes, every neighborhood of $p$ in $E$ is a neighborhood of $p$ in $K$, but not every neighborhood of $p$ in $K$ is a neighborhood of $p$ in $E$. You have to start with a neighborhood of $p$ in $K$ and show that that neighborhood of $p$ in $K$, not just... |
2,426,394 | <p>I'm sorry to bother you with this easy problem. But I'm working alone and totally confused.
It is the 1378th Problem from "Problems in Mathematical Analysis" written by Demidovich. The standard answer is
$$\frac{60(1+x)^{99}(1+6x)}{(1-2x)^{41}(1+2x)^{61}}$$</p>
<p>But when I followed the rutine
$$\frac{d(P(x)/Q(x))... | farruhota | 425,072 | <p>Alternatively:
$$[(1+x)^{100}(1-2x)^{-40}(1+2x)^{-60}]'=$$
$$100(1+x)^{99}(1-2x)^{-40}(1+2x)^{-60}+\\80(1+x)^{100}(1-2x)^{-41}(1+2x)^{-60}-\\120(1+x)^{100}(1-2x)^{-40}(1+2x)^{-61}=$$
$$20(1+x)^{99}(1-2x)^{-41}(1+2x)^{-61}\cdot \\ [5(1-2x)(1+2x)+4(1+x)(1+2x)-6(1+x)(1-2x)]=$$
$$\frac{20(1+x)^{99}}{(1-2x)^{41}(1+2x)^{6... |
1,044,005 | <p>so i'm trying to find the general formula of a sequence, $${18,30,46,66,90,118}$$</p>
<p>i found the first difference, $${12,16,20,24,24}$$</p>
<p>second difference, $${4,4,4,4}$$</p>
<p>If the second difference is 4, you start with $2n^2$</p>
<p>Then i have $${18,30,46,66,90,118}$$</p>
<p>-$${2,8,18,32,50,72}$... | turkeyhundt | 115,823 | <p>That's how I teach my students to try it. If the sequence's differences are constant, it's a linear progression. If the sequence's differences' differences are constant, start with $cn^2$ and work from there.</p>
|
1,991,600 | <p><a href="https://i.stack.imgur.com/2L9j9.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/2L9j9.jpg" alt="enter image description here" /></a></p>
<p>I have worked out the areas as <span class="math-container">$\pi/3$</span> for the circle and <span class="math-container">$2/\sqrt3$</span> for the ... | G Cab | 317,234 | <p>$$
\begin{gathered}
side_{\,tr} = 1\quad \quad h_{\,tr} = \frac{{\sqrt 3 }}
{2}\quad \quad r_{\,circ} = \frac{{\sqrt 3 }}
{6} \hfill \\
A_{\,circ} = \pi \frac{3}
{{36}} = \frac{\pi }
{{12}}\quad \quad A_{\,tr} = \frac{{\sqrt 3 }}
{4} \hfill \\
\rho = \frac{{A_{\,circ} }}
{{A_{\,tr} }} = \frac{\pi }
{{12}... |
799,602 | <p>If I have</p>
<pre><code>(¬q ⇒ r)
</code></pre>
<p>and use implication on this does it give me:</p>
<pre><code>(¬¬q v r)
</code></pre>
<p>from where I can then use double negation,</p>
<p>or does it give me </p>
<pre><code>¬(¬q v r)
</code></pre>
<p>I am thinking the former.</p>
<p>Thank you in advance.</p>... | Hector Pinedo | 30,533 | <p>You can use the book of Rotman "Advanced modern algebra" it covers the Brauer group of a field.</p>
<p>There is an article of Auslander and Goldman called "The Brauer group of a commutative ring," which generalizes the result of the book of Rotman to the context of commutative rings.</p>
|
75,704 | <p><strong>Bug introduced in 10.0 and fixed in 10.0.2</strong></p>
<hr>
<p>I was looking at integrals like:</p>
<pre><code>Integrate[HermiteH[50, x]*Exp[-x^2], {x, 0, Infinity}]
</code></pre>
<p>which gave me a "does not converge on $(0,\infty)$" error. On the other hand something like </p>
<pre><code>Integrate[(x... | Murta | 2,266 | <p>Well, not a PowerShell use, but an example of the new V10 function <code>RunProcess</code>.</p>
<p><a href="http://www.mls-software.com/opensshd-agent.html" rel="nofollow">Using this link</a>, I did this batch to add my private key in the repository ever time windows start:</p>
<pre><code>@ECHO OFF
for /f "tokens=... |
1,301,763 | <p>The only problem I have with this is knowing when you are solving for dx or dy. For example, this question which says</p>
<p>find the volume of the solid created by rotating the region bounded by y = 2x-4, y = 0, and x = 3, about the line x = 4. This one is dy. Instead of y = 2x-4, the person solved in respect to x... | randomgirl | 209,647 | <p><img src="https://i.stack.imgur.com/W73ym.png" alt="enter image description here">Sometimes it is easier to use horizontal strips than vertical (or the other way around). It also depends on the method you choose to find the volume, there is washer(disk) or shells. Shells you want parallel strips to the axis of rotat... |
2,093,661 | <p>In our algorithms class we defined the Fibonacci series:</p>
<p>$$F_0 = 0$$
$$F_1 = 1$$
$$F_{i+2} = F_i + F_{i+1}$$</p>
<p>After that we used that $F_{i+2} ≥ (\frac {1+\sqrt5} {2})^i$ but I can't see why that is true. Since it's the Fibonacci series I suspect that there should be a common known proof. If there isn... | Ben Grossmann | 81,360 | <p>Prove it inductively, noting that
$$
\left( \frac{1+\sqrt 5}{2}\right)^{i+2} = \left( \frac{1+\sqrt 5}{2}\right)^{i+1} + \left( \frac{1+\sqrt 5}{2}\right)^{i}
$$</p>
|
2,093,661 | <p>In our algorithms class we defined the Fibonacci series:</p>
<p>$$F_0 = 0$$
$$F_1 = 1$$
$$F_{i+2} = F_i + F_{i+1}$$</p>
<p>After that we used that $F_{i+2} ≥ (\frac {1+\sqrt5} {2})^i$ but I can't see why that is true. Since it's the Fibonacci series I suspect that there should be a common known proof. If there isn... | Doug M | 317,162 | <p>Not a prove as to why $F_{n+2} > (\frac {1+\sqrt 5}{2})^n$
But some insight why that expression is relevant.</p>
<p>Suppose, $\frac {F_{n+1}}{F_n}$ is converging to something.</p>
<p>$F_{n+1} = F_{n} + F_{n-1}\\
\frac {F_{n+1}}{F_n} = 1 + \frac {F_{n-1}}{F_n}$</p>
<p>So, supposing that this ratio is indeed con... |
54,815 | <p>A notorious question with prime numbers is estimating the gaps between consecutive primes. That is, if $(p_n)_{n \geq 1}$ is the canonical enumeration of the primes, then set $g_n = p_{n+1} - p_n$. It is shown that $g_n > \frac{c \log(n) \log \log(n) \log \log \log \log(n)}{(\log \log \log(n))^2}$ infinitely ofte... | Dan Brumleve | 2,003 | <p>How about <a href="https://en.wikipedia.org/wiki/Practical_number#Analogies_with_prime_numbers" rel="nofollow noreferrer">practical numbers</a>?</p>
<p>They aren't a superset of primes, but they are a "notion of 'almost primes'" as the title requests.</p>
<p>Hausman and Shapiro showed in 1984 that practical number... |
2,589,155 | <blockquote>
<p>Let $d$ be the standard metric on $\mathbb{R}$. i.e $d(x,y) = |x-y|$. Is there a distance $d'$ with $d'(x,y) \geq d(x,y)$ so that $(\mathbb{R},d')$ is compact?</p>
</blockquote>
<p>Is my proof correct?
Let $B_{\epsilon}^{d'}(x)$ denote a ball of radius $\epsilon$ centered on $x$ with respect to $d'$.... | José Carlos Santos | 446,262 | <p>Your proof is correct, <em>assuming</em> that $(\mathbb{R},d)$ isn't totally bounded. Do you know how to prove that?</p>
|
3,994,280 | <p>Show that <span class="math-container">$(\lnot(p\lor(\lnot p\land q))$</span> is logically equivalent to <span class="math-container">$\lnot p \land \lnot q$</span>.</p>
<p>I am wondering what I did is correct. Very new to learning simple logic.</p>
<p><span class="math-container">$$(\lnot(p\lor(\lnot p\land q)) \eq... | amWhy | 9,003 | <p>This is an alternative rount, and corrects for your misuse of DeMorgan's in the last step.</p>
<p><span class="math-container">$$(\lnot(p\lor(\lnot p\land q))) \equiv \lnot p \land \lnot(\lnot p \land q)\tag{DeMorgan's}$$</span>
<span class="math-container">$$\equiv \lnot p \land (p \lor \lnot q)\tag{DeMorgan's}$$</... |
90,006 | <p>I have two problem collections I am currently working through, the "Berkeley Problems in Mathematics" book, and the first of the three volumes of Putnam problems compiled by the MAA. These both contain many problems on basic differential equations.</p>
<p>Unfortunately, I never had a course in differential equation... | GEdgar | 442 | <p>For example, show the limit is $\le 1/2$ like this:</p>
<p>$$
\frac{n+1}{2n+3} < \frac{n+1}{2n+2} = \frac{1}{2}
$$</p>
|
1,842,448 | <p>$F(x)= 2x^2 -3x$. </p>
<p>find the range of $x$ to check whether the function the is strictly increasing and strictly decreasing.</p>
| user190080 | 190,080 | <p>The function
$$
F(x)= 2x^2 -3x
$$
is strictly increasing (respectively decreasing) for
$$
f(x):=F'(x)=4x-3>0 \text { respectively } f(x)=4x-3<0 \tag1
$$
this is a linear function which is strictly increasing itself on whole $\mathbb R$. Just observe that $$ f(x)'=4>0$$ holds. Now we just calculate the roo... |
2,127,110 | <p>I am working on this differetiation problem:</p>
<p>$ \frac{d}{dx}x(1-\frac{2}{x})$</p>
<p>and I am currently stuck at this point:</p>
<p>$1\cdot \left(1-\frac{2}{x}\right)+\frac{2}{x^2}x$</p>
<p>Symbolab tells me this simplifies to $1$ but I do not understand how. I am under the impression that;</p>
<p>$1\cdot... | 5xum | 112,884 | <p>$$1\cdot (1-\frac2x) + \frac{2}{x^2}x = 1-\frac2x + \frac2x = 1$$ for all $x\neq 0$.</p>
<p>I have no idea where you got $x^{-x^2}$ from...</p>
|
588,228 | <p>A,B,C are distinct digits of a three digit number such that </p>
<pre><code> A B C
B A C
+ C A B
____________
A B B C
</code></pre>
<p>Find the value of A+B+C.</p>
<p>a) 16 b) 17 c) 18 d) 19</p>
<... | souparno | 397,876 | <p>I am answering the edited question without trial and error method as it will consume more time. we can see that A+B must be 10 as C added to it will result in C in the right most digit and the carry will be 1(carry cannot be 2 as A+B cannot be 20,as they r single digit number).now for the middle column B+C+A+1(i.e c... |
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