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 |
|---|---|---|---|---|
1,422,666 | <p>I've just run into this problem, and was able to go as far, and understand the induction step up to the bolded section. The last part I found in the back of my book, italicized, I can't understand.</p>
<p>Use induction to prove: $n^2\geq 2n + 1$ for all $n\in \mathbb{N}$ and $n\geq 3$.</p>
<p>Base: $3^2\geq 6+1$.<... | 2'5 9'2 | 11,123 | <p>You have <em>assumed</em> that $\color{blue}{k^2}\geq\color{#af5500}{2k+1}$ for some specific $k$. So $$\begin{align}(k+1)^2&=\color{blue}{k^2}+2k+1\\
&\geq\color{#af5500}{2k+1}+2k+1\\&=2(k+1)+2k\\&\geq2(k+1)+1\end{align}$$ where the last inequality is because $2k>1$.</p>
|
4,621,227 | <p>This is a very soft and potentially naive question, but I've always wondered about this seemingly common phenomenon where a theorem has some method of proof which makes the statement easy to prove, but where other methods of proof are incredibly difficult.</p>
<p>For example, proving that every vector space has a ba... | Qiaochu Yuan | 232 | <p>Very often the first proof of a result which appears in the literature is extremely messy because the mathematician who proved it is working at the very edge of what is possible with the tools of the day; then it gets simplified over time as other mathematicians better understand what is going on and develop better ... |
486,239 | <p>I was really stuck and tried many times to differentiate the following series, and tried to convince myself that the differential form of a triangular wave is the square wave.</p>
<p>But I couldn't work it out as I found those sins and cos dont match up</p>
<p>Square wave has this form </p>
<p><img src="https://i... | Anthony Carapetis | 28,513 | <p>Start with some graphical reasoning. When you plot these square and triangular waves, you'll notice that they need to be $\pi/2$ out of phase in order for the square wave to match up with the slope of the triangular wave; there will also be some vertical scaling you need to apply. This phase shift will explain why t... |
486,239 | <p>I was really stuck and tried many times to differentiate the following series, and tried to convince myself that the differential form of a triangular wave is the square wave.</p>
<p>But I couldn't work it out as I found those sins and cos dont match up</p>
<p>Square wave has this form </p>
<p><img src="https://i... | Ross Millikan | 1,827 | <p>The key observation is that a sine wave is the same as a cosine wave, but shifted by $\frac \pi 2$ As the triangle wave is odd, the derivative of the square wave is even (plot it) so should be a sum of cosines.</p>
|
237,960 | <p>How could I solve this problem?</p>
<blockquote>
<p>Find the first digit of $2^{4242}$ without using a calculator.</p>
</blockquote>
<p>I know how to find the last digit with modular arithmetic, but I can't use that here.</p>
| picakhu | 4,728 | <p>Presenting an alternate method (no logs, but needs the knowledge that doubling time is ~ $70$/rate) </p>
<p>Starting with </p>
<p>$$2^{10}=1024$$
which is
$$1000 \times 1.024$$ or a 2.4% increase. </p>
<p>Then,
$$70/2.4 \approx 29$$
implies that
$$2^{290}\sim 2 \times 10^k$$
for some k. </p>
<p>Then, $$2^{... |
3,783,186 | <p>I am trying to prove that <span class="math-container">$$2≤\int_{-1}^1 \sqrt{1+x^6} \,dx ≤ 2\sqrt{2} $$</span> I learned that the equation <span class="math-container">$${d\over dx}\int_{g(x)}^{h(x)} f(t)\,dt = f(h(x))h'(x) - f(g(x))g'(x) $$</span> is true due to Fundamental Theorem of Calculus and Chain Rule, a... | mathguy23123 | 367,767 | <p>So there are two inequalities to be proved. You can use that <span class="math-container">$\sqrt{1+x^6} \leq \sqrt{2}$</span> for all <span class="math-container">$x \in [-1,1]$</span> for the upper bound, as it follows <span class="math-container">$\int_{[-1,1]} \sqrt{1+x^6} dx\leq \int_{[-1,1]} \sqrt{2} dx\leq 2 \... |
2,709,878 | <p>How do I know that an equation will have an extraneous solution? </p>
<p>For example in this question:
2log9(x) = log9(2) + log9(x + 24)</p>
| lab bhattacharjee | 33,337 | <p>First identify where each of the terms remains defined.</p>
<p>For real domain, we need $x,x+24>0\implies x>0$</p>
<p>$$\implies\log_9(x^2)=\log_92+\log_9(x+24)=\log_92(x+24)$$</p>
<p>$$\iff x^2=2(x+24)$$</p>
|
4,474,095 | <p>There is one thing I can't grasp about the proof given in the Linear Algebra Done Right book by Sheldon Axler (attached below).</p>
<p>In the last part it says that <span class="math-container">$(T - \lambda_1I)...(T - \lambda_mI)v = 0$</span>, hence <span class="math-container">$T - \lambda_jI$</span> is not inject... | CyclotomicField | 464,974 | <p>If <span class="math-container">$\lambda_j$</span> is the only eigenvalue then <span class="math-container">$(T−\lambda_j)^n$</span> is the full expression and reordering doesn't change anything.</p>
|
123,712 | <p>How can I prove that the function
$$ f(x) = \left\{\begin{array}{l l} x &\text{if }x \in \mathbb{Q} \\ -x & \text{if } x \notin \mathbb{Q}
\end{array} \right. $$
is discontinuous for $x \neq 0$, using $\epsilon$'s and $\delta$'s?</p>
<p>I see that is truth. But I cannot prove using only $\eps... | Davide Giraudo | 9,849 | <p>Let $x>0$, and assume that we can find a $\delta>0$ such that if $|y-x|\leq\delta$ then $|f(x)-f(y)|<x/2$. If $x$ is rational, it means $|x-f(y)|<x/2$ so $x/2<f(y)<3x/2$ for all $y\in (x-\delta,x+\delta)$. In particular, $y\in\mathbb Q$ for all $y\in (x-\delta,x+\delta)$.
If $x$ i irrational, then... |
1,344,001 | <p>I need to solve an expression of this kind (solve for $x$):</p>
<p>$e^{\pi i x} -e^{-\pi ix} = 2yi$</p>
<p>Both $x$ and $y$ are real numbers, $y$ is given. I have no clue on how to solve it analytically.</p>
<p>All I know is that I can rewrite this as:</p>
<p>$\sin(x\pi) = y$</p>
<p>so:</p>
<p>$x=\frac{\arcsin... | corindo | 40,305 | <p>$\sin(x\pi) = y$. And you have </p>
<p>$$ x = \frac{(-1)^{k+1} \arcsin y + k\pi}{\pi} = \frac{(-1)^{k+1} \arcsin y}{\pi} + k, \quad k \in \mathbb{Z} $$</p>
|
2,430,529 | <p>How should I define a vector, that has equal angles to vectors $\vec{i}, \vec{i} + \vec{j}$ and $\vec{i} + \vec{j} + \vec{k}$?</p>
<p>After looking at the problem in a graphical way, I tried taking average from $\vec{i}$ and $\vec{i} + \vec{j} + \vec{k}$, modifying the $\vec{i}$ vector to be of same length than the... | user247327 | 247,327 | <p>The length is, of course, irrelevant to the angles. Take the vector to be ai+ bj+ ck. The angle it makes with i is $cos^{-1}\left(\frac{a}{\sqrt{a^2+ b^2+ c^2}}\right)$. The angle it makes with i+ j is $cos^{-1}\left(\frac{a+ b}{\sqrt{2}/sqrt{a^2+ b^2+ c^2}}\right)$. The angle it makes with i+ j+ k is $cos^{-1}\... |
416,407 | <blockquote>
<p>What examples are there of habitual but unnecessary uses of the axiom of
choice, in any area of mathematics except topology?</p>
</blockquote>
<p>I'm interested in standard proofs that use the axiom of choice, but where
choice can be eliminated via some judicious and maybe not quite obvious
rephrasing. ... | Ege Erdil | 131,052 | <p>I've seen Tychonoff's theorem be used to prove that the <span class="math-container">$ p $</span>-adic integers are compact. The proof is easy: there is a natural embedding</p>
<p><span class="math-container">$$ \mathbb Z_p \to \prod_{k=1}^{\infty} (\mathbb Z/p^k \mathbb Z) $$</span></p>
<p>whose image is closed, an... |
416,407 | <blockquote>
<p>What examples are there of habitual but unnecessary uses of the axiom of
choice, in any area of mathematics except topology?</p>
</blockquote>
<p>I'm interested in standard proofs that use the axiom of choice, but where
choice can be eliminated via some judicious and maybe not quite obvious
rephrasing. ... | Samuel Mimram | 39,955 | <p><a href="https://arxiv.org/abs/math/0605779" rel="nofollow noreferrer">You can divide sets by two</a> (or more): in classical ZF, if <span class="math-container">$A\sqcup A\simeq B\sqcup B$</span> then <span class="math-container">$A \simeq B$</span>.</p>
|
1,481,106 | <p>I am having a difficult time solving this problem. I have tried this several different ways, and I get a different result, none of which is correct, every time. I've derived an answer geometrically and cannot replicate it with a double integral.</p>
<p>Here's the problem: Use a double integral to find the area betw... | E.H.E | 187,799 | <p>by using the Cartesian coordinates
$$\int_{-\frac{\sqrt{15}}{2}}^{\frac{\sqrt{15}}{2}}\int_{1-\sqrt{4-y^2}}^{\sqrt{4-y^2}}dxdy$$ </p>
|
2,306,122 | <p>Show $X=\{n \in \mathbb{N}: \text{n is odd and} \ n = k(k+1) \text{for some} \ k \in \mathbb{N}\}=\emptyset$</p>
<p>My proof is as follow, please point if I have made any mistake. </p>
<p><strong>proof:</strong></p>
<p>we have $\emptyset \subseteq X$
suppose $X≠\emptyset$ pick $n \in X$</p>
<p>Then there are 2 ... | avs | 353,141 | <p>An even shorter proof (not that yours is wrong): </p>
<p>One of $k, k+1$ must be even, therefore so is the product $k(k+1)$.</p>
|
2,645,406 | <p>I am reading Rudin's real and complex analysis. On page 14 we see $\underset{n} {\text{sup }} f_n$, and on page 15 we see max$\{f,g\}$. These two are obviously different. I searched some answers and found an example: </p>
<p>sup $\{f,g\} = f(x)$ wehn $f(x)\geq g(x)$ and $g(x)$ when $f(x) < g(x)$ </p>
<p>But the... | Jay Zha | 379,853 | <p>If $x = \max A$ for some set $A \subset \mathbb R$, then $x \ge y$ for any $y \in A$, and $x \in A$.</p>
<p>But the $\sup A$ does not need to be in $A$. By definition if $x = \sup A$, for some set $A \subset \mathbb R$, then $ x \ge y$ for any $y \in A$, but $x \le y$ if $y$ is an upper bound of $A$.</p>
|
248,325 | <p>is there some connection between a curve in the algebraic geometry sense, e.g.</p>
<blockquote>
<p>Separated scheme of finite type over spec($k$)</p>
</blockquote>
<p>for a field $k$</p>
<p>and a curve in the sense of a smooth map from an interval in $\mathbb R$ to $\mathbb R^n$?</p>
| Bin Wang | 88,343 | <p>I think when you mention a smooth map, the condition seems appear in the category of diffential geometry, which is quite different. The concept of 1 dimensional separated scheme (since we consider curves) is in fact a generalization of algebraic curves(not necessarily smooth). More precisely, an algebraic curve is a... |
4,417,353 | <p><a href="https://i.stack.imgur.com/td05G.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/td05G.png" alt="Here is the question. I sent it as a image since I could not find how to write a1 as below." /></a></p>
<p>My first approach is that I need to use induction on that. However the problem is that... | Hagen von Eitzen | 39,174 | <p>Once you know <em>that</em> the sequence converges, what can you infer about the actual <em>value</em> of <span class="math-container">$a:=\lim a_n$</span>?
Since <span class="math-container">$f\colon x\mapsto \sqrt2^x$</span> is continuous on <span class="math-container">$\Bbb R$</span>, we must have <span class="m... |
972,191 | <p>Consider the linear transformation of $\mathbb{R}^3$ given by $Ax = (a \cdot x)a+ |a|^2x$. Is $A$ symmetric? Is it positive?</p>
<p>I know that a matrix is symmetric IFF $(Ax,y) = (x,Ay)$ and positive if $x \cdot Ax > 0$ for all vectors $x \not = 0$.</p>
| Timbuc | 118,527 | <p>Take the usual, canonical basis of $\;\Bbb R^3\;$ :</p>
<p>$$u_1=\begin{pmatrix}1\\0\\0\end{pmatrix}\;,\;\;u_2=\begin{pmatrix}0\\1\\0\end{pmatrix}\;,\;\;u_3=\begin{pmatrix}0\\0\\1\end{pmatrix}$$</p>
<p>and suppose $\;a=\begin{pmatrix}a_1\\a_2\\a_3\end{pmatrix}\;$ , so that</p>
<p>$$\begin{align*}&Au_1:=a_1a+(... |
972,191 | <p>Consider the linear transformation of $\mathbb{R}^3$ given by $Ax = (a \cdot x)a+ |a|^2x$. Is $A$ symmetric? Is it positive?</p>
<p>I know that a matrix is symmetric IFF $(Ax,y) = (x,Ay)$ and positive if $x \cdot Ax > 0$ for all vectors $x \not = 0$.</p>
| Yiorgos S. Smyrlis | 57,021 | <p>Checking symmetry of $A$:
$$
( y,Ax)= ( y,(a \cdot x)a+ |a|^2x)=(a\cdot x)(a\cdot y)+\lvert a\rvert^2(x,y)=( x,Ay)=(Ay,x).
$$
Hence $A$ is indeed symmetric.</p>
<p>Positive definite
$$
( x,Ax)=(a\cdot x)^2+\lvert a\rvert^2(x,x)\ge \lvert a\rvert^2\lvert x\rvert^2,
$$
hence it is non-negative definite, and if $a\ne ... |
3,841,535 | <p>I'm having a lot of trouble solving this question via the differentiating with respect to a parameter method. I can get the correct result for the integral containing sine, but I'm totally lost when it comes to evaluating the integral containing cosine. Here's the problem statement:</p>
<p>Given:</p>
<p><span class=... | Aadhaar Murty | 826,105 | <p>As suggested by @Abhi, differentiating w.r.t. <span class="math-container">$k$</span> will give you a direct answer. We have,</p>
<p><span class="math-container">$$\frac {d}{dk} \left(\frac {k}{a^{2}+ k^{2}}\right) = \int_{0}^{\infty} e^{-ax} \cdot \partial_{k}\sin(kx) dx = \int_{0}^{\infty}xe^{-ax} \cos(kx) dx = \b... |
674,621 | <p>I am trying to figure out what the three possibilities of $z$ are such that </p>
<p>$$
z^3=i
$$</p>
<p>but I am stuck on how to proceed. I tried algebraically but ran into rather tedious polynomials. Could you solve this geometrically? Any help would be greatly appreciated.</p>
| Gilles 'SO- stop being evil' | 1,853 | <p>You can solve this geometrically if you know <a href="http://en.wikipedia.org/wiki/Polar_coordinates" rel="noreferrer">polar coordinates</a>.</p>
<p>In polar coordinates, multiplication goes $(r_1, \theta_1) \cdot (r_2, \theta_2) = (r_1 \cdot r_2, \theta_1 + \theta_2)$, so cubing goes $(r, \theta)^3 = (r^3, 3\theta... |
1,601,427 | <blockquote>
<p>Let $a,b,c$ be three nonnegative real numbers. Prove that $$a^2+b^2+c^2+3\sqrt[3]{a^2b^2c^2} \geq 2(ab+bc+ca).$$</p>
</blockquote>
<p>It seems that the inequality $a^2+b^2+c^2 \geq ab+bc+ca$ will be of use here. If I use that then I will get $a^2+b^2+c^2+3\sqrt[3]{a^2b^2c^2} \geq ab+bc+ca+3\sqrt[3]{a... | ki3i | 202,257 | <p>Yet another way in which this can be shown using <a href="https://en.wikipedia.org/wiki/Schur%27s_inequality">Schur's inequality</a> in tandem with the AM-GM inequality is as follows:
$$
a^2+b^2+c^2+3(a^2b^2c^2)^{1/3}\geqslant a^{2/3}b^{4/3} + a^{4/3}b^{2/3} +b^{2/3}c^{4/3} + b^{4/3}c^{2/3} + a^{2/3}c^{4/3} + a^{4/3... |
3,828,205 | <blockquote>
<p>Find all integers <span class="math-container">$x$</span> and <span class="math-container">$y$</span> that satisfy <span class="math-container">$$x^4-12x²+x^2y^2+30 < 0$$</span></p>
</blockquote>
<p>Letting <span class="math-container">$a = x^2$</span> and <span class="math-container">$b = y^2$</span... | markvs | 454,915 | <p>You have <span class="math-container">$(a-6)^2< 6$</span>, <span class="math-container">$a\ge 0$</span>. So <span class="math-container">$(a-6)^2\in \{0,1,4\}$</span>. If <span class="math-container">$(a-6)^2=0$</span>, then <span class="math-container">$a=6$</span>, impossible since <span class="math-container">... |
1,165,828 | <p>I run into a problem when I'm trying to prove how $\tan^2x+1 = \sec^2x$, and $1+\cot^2x=\csc^2x$</p>
<p>I understand that $\sin^2x+\cos^2x = 1$. (To my understanding 1 is the Hypotenuse, please correct me if I'm wrong). If referring to a Pythagorean triangle, let's say a triangle where $a=3$, $b=4$, and $c=5$, or $... | turkeyhundt | 115,823 | <p>The first identity works because $\frac{3}{5}^2+\frac{4}{5}^2=\frac{5}{5}^2$. When thinking of these trig ratios, you have to think of them in terms of both triangle sides you are interested in. Not merely one leg being equal to $\sin$, for example.</p>
<p>The beauty of the "unit" circle is that with the hypotenu... |
1,756,567 | <p>Let A be a 3*3 matrix and $A^{2014}=0$. Must $A^3$ be the zero matrix? I can work out that I-A is invertible, but I don't know how to proceed further.</p>
| Michael Burr | 86,421 | <p>This is very much overkill for this problem (Wojowu's suggestion in the comments is a very nice and clean solution). This answer classifies the possible $A$'s.</p>
<p>Since $A^{2014}=0$, all of the eigenvalues of $A$ must be $0$ (for if $\lambda\not=0$ were an eigenvalue with eigenvector $v$, then $A^{2014}v=\lamb... |
1,565,406 | <p>Which interpolation method should I use for complicated "smooth" curves such as </p>
<p>$\frac{sin(x)}{x}$ for $x>0$.</p>
| bubba | 31,744 | <p>The approximation method should generally be tuned to the function (or at least the type of function) that you're trying to approximate. There are general-purpose one-size-fits-all methods, but a specialized method will typically be much better.</p>
<p>If I had to choose one particular class of methods, I suppose i... |
269,062 | <p>Consider the following integral
$$
pv\int_0^{\infty}e^{N(-2Ax+A\log x)}\frac{e^{-B\log x}}{1-2x}dx
$$
where $A,B>0$ and we take the Cauchy principal value at $x=1/2$. I am interested in obtaining the asymptotics when $N$ is very big. The first thing I thought of was some variant of Laplace's method but I am unsur... | Fedor Petrov | 4,312 | <p>You may estimate the integral on $[1,\infty)$ somehow (it does not affect the asymptotics), after that consider the integral of $\int_0^{1/2}f_N(1/2-t)+f_N(1/2+t)dt$, where $f_N$ is your function. This already does not have a singularity at 0 and you may apply Laplace method. </p>
|
194,312 | <p>I'm preparing myself to a combinatorics test. A part of it will concentrate on the pigeonhole principle. Thus, I need some hard to very hard problems in the subject to solve.
I would be thankful if you can send me links\books\or just a lone problem.</p>
| Holdsworth88 | 22,437 | <p><a href="http://www.math.auckland.ac.nz/~olympiad/Training/Numbers/Pigeonhole.pdf" rel="nofollow">This</a> turned up in a routine google search of the phrase "pidgeonhole principle exercise" and appears to be training problems for the New Zealand olympiad team. It contains numerous problems and has some solutions in... |
3,719,575 | <p>For <span class="math-container">$\lambda =0$</span> and <span class="math-container">$\lambda <0$</span> the solution is the trivial solution <span class="math-container">$x\left(t\right)=0$</span></p>
<p>So we have to calculate for <span class="math-container">$\lambda >0$</span></p>
<p>The general solution... | Disintegrating By Parts | 112,478 | <p>There only exist solutions for specific values of <span class="math-container">$\lambda$</span>, and those <span class="math-container">$\lambda$</span> are the eigenvalues of the underlying problem. So the goal of finding non-trivial solutions of the following equations is to (a) find the <span class="math-containe... |
1,453,682 | <p>I'm having trouble classifying the solution set of the systems $Ax = b$ and $Ax = 0$.</p>
<p>Let A be an ($m \times n$) matrix.</p>
<p><strong>Case I:</strong></p>
<p><em>For $Ax = 0$, the system has a unique solution (the trivial one) when A is invertible, and infinitely many solutions when A is not.</em> </p>
... | Bernard | 202,857 | <p>In case II, note that in general, $A$ will not be invertible, as this requires $m=n$.</p>
<p>The condition for having solutions is the rank of the augmented matrix is equal to $\operatorname{rank}A$ (in all cases it is $\ge\operatorname{rank}A$). It is the case if $A$ has rank $m$, which implies $n\ge m$ and $A$ i... |
742,160 | <p>Answer true or false to each of the following questions. If a statement is true, prove it. If a statement is false, give a counterexample.</p>
<ol>
<li>For all sets $A$,$B$ and $C$: IF $A ⊆ B$ and $A ⊆ C$, Then $A ⊆ (B ∩ C)$</li>
<li>For all sets $A$ and $B$, if $|A| \le |B|$, then $A ⊆ B$</li>
</ol>
| hmakholm left over Monica | 14,366 | <p>An injection $A\to B$ provides a correspondence between $A$ and some subset of $B$ -- that, is an INjection points to a copy of $A$ INside $B$.</p>
|
1,027,707 | <p>In how many ways can 100 identical chairs be divided among 4 different rooms so that each room will have 10,20,30,40 or 50 chairs?</p>
<p>I'm having problems coming up with the generating function for this question. </p>
<p>The answer given is 68. </p>
| Qiaochu Yuan | 232 | <p>You are correct that the category of finite-dimensional vector spaces is not cartesian closed. However, it is <a href="http://ncatlab.org/nlab/show/closed+monoidal+category" rel="nofollow">closed monoidal</a>: this is like being cartesian closed, but with the cartesian product replaced with an arbitrary monoidal pro... |
35,877 | <p>I got stucked while I was working out whether
$$\frac{4n-3}{n+43}$$
converges. I would be pleased if I could a hint of the above question.</p>
| Dennis Gulko | 6,948 | <p>What are the tools that you can use? One possible way is to use Cauchy's theorem: Show that for any given $\varepsilon>0$ there exists $N>0$ such that for all $n>N$ we have $\left|\frac{4n-3}{n+43}-4\right|<\varepsilon$</p>
|
2,783,224 | <p>$$\lim_{x \rightarrow 0}\frac{x^2 \sin(Kx)}{x - \sin x} =1$$</p>
<p>Here K is a constant whose value I want to find,</p>
<p>I got it by writing the series expansion of $\sin(\theta)$, but couldn't by L'hospital rule or standard limits,</p>
<p>This is what I tried:</p>
<p>$$\lim_{x \rightarrow 0}\frac{x^2 \sin(Kx... | Dave | 334,366 | <p>L'Hopital's rule certainly works: you can apply L'Hopital's rule three times to get the limiting function to be continuous at $x=0$.</p>
<p>The third derivative of the numerator is $$-k(6kx\sin(kx)+(k^2x^2-6)\cos(kx))$$ and the third derivative of the denominator is $\cos(x)$. Hence, after three applications of L'H... |
540,029 | <p>I know that for any matrix <span class="math-container">$A$</span>, <span class="math-container">$AA^{\ast}$</span> is positive semidefinite (where <span class="math-container">$A^\ast$</span> is <span class="math-container">$\overline{A}^T$</span>). Please help me show the following statement</p>
<blockquote>
<p... | copper.hat | 27,978 | <p>If $A \ge 0$ and $A = A^*$, then $A$ is unitarily diagonalizable and all eigenvalues are real and non-negative. That is, for some unitary $U$, we have $U^*AU = \Lambda$, where $\Lambda = \operatorname{diag}(\lambda_1,...,\lambda_n)$, and $\lambda_k \ge 0$.</p>
<p>If we let $\Lambda^{\frac{1}{2}} = \operatorname{dia... |
1,277,827 | <p>I came across a question from Munkres:
Problem 24.11: If A is a connected subspace of X, does it follow that IntA and BdA are connected? Does the converse hold?</p>
<p>Thinking about the converse, I know that in general, if Bd A is connected, it doesn't necessarily mean that A is connected (e.g. the rationals). But... | Andreas Blass | 48,510 | <p>Suppose that $A$ is closed in $X$, that $X$ and Bd$(A)$ are connected, but (toward a contradiction) that $A$ is disconnected. So $A$ is the union of two disjoint, nonempty subsets $P$ and $Q$ that are closed in $A$ and therefore in $X$. Now consider any point $x$ in Bd$(A)$. Since $A$ is closed, it follows that $x\... |
261,474 | <p>Let <span class="math-container">$A$</span> be the matrix <span class="math-container">$$A = \begin{pmatrix} 1 & \sqrt{2}\\ -\sqrt{2} & -1\\ \end{pmatrix}$$</span></p>
<br>
<blockquote>
<p>Compute the matrix <span class="math-container">$B = 3A -2A^2 - A^3 -5A^4 + A^6$</span>.</p>
</blockquote>
<p>Could any ... | P.. | 39,722 | <p>In this example probably the best way is to use Raymond Manzoni's hint.</p>
<p>In general if $A$ is similar to $C$ with $A=P^{-1}CP$ for some invertible $P$ then for any polynomial $\phi(x)$: $$\phi(A)=P^{-1}\phi(C)P.$$ In your case use that $A$ is similar to $\begin{pmatrix}1+\sqrt{2}&0\\0&1-\sqrt{2}\end{... |
261,474 | <p>Let <span class="math-container">$A$</span> be the matrix <span class="math-container">$$A = \begin{pmatrix} 1 & \sqrt{2}\\ -\sqrt{2} & -1\\ \end{pmatrix}$$</span></p>
<br>
<blockquote>
<p>Compute the matrix <span class="math-container">$B = 3A -2A^2 - A^3 -5A^4 + A^6$</span>.</p>
</blockquote>
<p>Could any ... | Martin Sleziak | 8,297 | <p>We have the characteristic polynomial $ch_A(x)=\begin{vmatrix}1-x&\sqrt{2}\\-\sqrt{2}&1-x\end{vmatrix}=(x-1)^2+2=x^2-2x+3$.</p>
<p>Then by <a href="http://en.wikipedia.org/wiki/Cayley-Hamilton_theorem" rel="nofollow">Cayley-Hamilton theorem</a> we know that $A^2-2A+3=0$.</p>
<p>So if you divide your polyno... |
3,471,585 | <blockquote>
<p>All roots of polynomial <span class="math-container">$x^3+ax^2+17x+3b$</span>, <span class="math-container">$a,b\in \Bbb Z$</span> are integers. Prove that this polynomial doesn't have same roots.</p>
</blockquote>
<p>My plan was to find all three solutions and then compare them.</p>
<p>I don't know... | Bill Dubuque | 242 | <p><strong>Hint</strong> <span class="math-container">$ $</span> the roots are distinct <span class="math-container">$\bmod 3\!:\ f \equiv x(x^2\!+\!ax\!-1)\not \equiv x(x\!-\!r)^2\,$</span> by <span class="math-container">$\,r^2\not\equiv 1$</span></p>
|
1,251,334 | <blockquote>
<p>$||x-2|-3| >1$, then $x$ belongs to:<BR> (a) $(-\infty, -2) \cup (0,4) \cup(6,\infty)$<br> (b) $(-1,1)$<br> (c) $(-\infty, 1)\cup (1,
> \infty)$ <br> (d) $(-2,2)$</p>
</blockquote>
<p><strong><em>Answer:(a)</em></strong></p>
<p>My solution:</p>
<p>let $|x-2| = p,$<br>
$|p-3|>1$ and finally... | Simon S | 21,495 | <p>$||x−2|−3|>1$ then either $|x-2| - 3 > 1$ or $|x-2| - 3 < -1$; respectively</p>
<p>$$|x-2| > 4 \ \text{ or } |x-2| < 2$$</p>
<p>The first of these is equivalent to $x - 2 > 4$ or $x - 2 < -4$; i.e., $$x > 6 \text{ or } x < -2$$</p>
<p>The second is equivalent to $-2 < x - 2 < 2$; ... |
2,933,577 | <p>Eliminate the parameters to find a Cartesian equation of the curve and sketch the curve.</p>
<p><span class="math-container">$x = e^t – 1, y = e^{2t}$</span>.</p>
<p>My attempt:</p>
<p><span class="math-container">$x = e^{t} - 1$</span></p>
<p><span class="math-container">$x + 1 = e^{t}$</span></p>
<p><span cla... | Mike | 544,150 | <p>How you would sketch this:</p>
<p>Hint 1: Note that <span class="math-container">$x$</span> can take on each possible value in precisely the set <span class="math-container">$(-1,\infty)$</span> as <span class="math-container">$t$</span> varies over the reals.</p>
|
1,879,149 | <p>Let $\mu$ be a Radon measure on $\mathbb{R}$. (<em>i.e.</em>, a locally finite Borel measure)</p>
<p>Let $I$ be an interval of $\mathbb{R}$.</p>
<p>Let $\gamma : I \mapsto \mathbb{R}$ be a <strong>monotone</strong> function that is Lebesgue-Stieltjes integrable with respect to $\mu$:
$$\int_I \gamma(x) \mathrm{d}\... | PhilippeC | 315,538 | <p>@Luiz Cordeiro: thanks for your answer.</p>
<p>I may have an alternative answer for the case $I = [a, b]$, with is reminiscent of the good old Riemann integration and does not use Lusin theorem.
Assume $\gamma(a) \neq \gamma (b)$ (else, there is nothing to do).</p>
<p>Let $\epsilon > 0$. Define $\tilde\epsilon ... |
28,258 | <p>The <a href="https://math.stackexchange.com/questions/tagged/set-theory" class="post-tag" title="show questions tagged 'set-theory'" rel="tag">set-theory</a> tag is explicitly about mathematics (and metamathematics) in the context of $\mathsf{ZFC}$ and its subsystems and extensions.</p>
<p>Every now and the... | Martin Sleziak | 8,297 | <p>It seems that there is a disagreement whether or not the new tag <a href="https://math.stackexchange.com/questions/tagged/alternative-set-theories" class="post-tag" title="show questions tagged 'alternative-set-theories'" rel="tag">alternative-set-theories</a> should be used in conjunction with <a href="http... |
6,825 | <p>Consider four <span class="math-container">$n$</span>-sided polygons arranged so that their centers are the vertices of a square. The square is exactly as large as the diameter of the smallest circle enclosing each polygon. The polygons do not overlap. Polygons are oriented so that each one has either exactly two si... | Paul VanKoughnett | 2,215 | <p>There are two clear cases. When $2n$ is divisible by $4$, each polygon will have a horizontal side and a vertical side touching other polygons. So the number of sides each polygon contributes to the interior is just the number of sides between a horizontal and a vertical side. (Each side of the "interior" polygon... |
95,616 | <p>I have tried to adapt <a href="https://mathematica.stackexchange.com/a/16290/29734">this answer</a> to my problem of calculating some bosonic commutation relations, but there are still some issues.</p>
<p>The way I'm implementing the commutator is straightforward:</p>
<pre><code>commutator[x_, y_] :=x**y-y**x
</co... | QuantumDot | 2,048 | <p>The function <code>NonCommutativeMultiply</code> has too long of a name, so I make a short-hand version of it (<code>NCP</code> stands for <strong>n</strong>on-<strong>c</strong>ommutative <strong>p</strong>roduct):</p>
<pre><code>NCP[x___] := NonCommutativeMultiply[x];
</code></pre>
<p>Now, here's the code</p>
<... |
171,449 | <p><strong>Theorem</strong> Every sequence {$s_n$} has a monotonic subsequence whose limit is equal to $\limsup s_n$. I think to show that there exist a monotonic subsequence is kind of straight forward but I could show that there exist such subsequences whose limit is $\limsup s_n$.</p>
| vanguard2k | 35,711 | <p><br></p>
<p>You could plot the function $e^{rt}$ for different values of $t$ with $r$ being your annual exponential growth rate. Then you could evaluate this function at $t=10$ or $t=1000$. <br> Is this what you were looking for?</p>
|
3,155,241 | <p>Let <span class="math-container">$\mathbb{A}^n$</span> denote the set of n-tuples of elements from field <span class="math-container">$k$</span> and <span class="math-container">$I(X)$</span> the ideal of polynomials in <span class="math-container">$k[x_1,...,x_n]$</span> that vanish every point in <span class="math... | dcolazin | 654,562 | <p>If <span class="math-container">$k$</span> is finite, you can build polynomials what vanish in all <span class="math-container">$\mathbb{A}^n$</span> thanks to <span class="math-container">$p(x) = \Pi_{z\in k}(x-z)$</span>.</p>
<p>If <span class="math-container">$k$</span> is infinite, then <span class="math-contai... |
2,243,900 | <p>I've researched this topic a lot, but couldn't find a proper answer to this, and I can't wait a year to learn it at school, so my question is:</p>
<blockquote>
<p>What exactly is calculus? </p>
</blockquote>
<p>I know who invented it, the Leibniz controversy, etc., but I'm not exactly sure what it is. I think I ... | LDBerriz | 299,917 | <p>The essence of Calculus is the consideration of quantities that are very small such that they are almost zero but not quite. Doing so allows for the calculation of things that occur in an instant, whether an instant of time, or an instant of any dimension. These quantities usually are infinitely small and when they... |
2,218,960 | <blockquote>
<p>Let $(W_t,\mathscr{F_t})$ be a Wiener process and let
$$M_t=M_0e^{W_t-t/2}\qquad t\ge0$$
where $M_0$ is deterministic.
Show that, for $\epsilon>0$, $\tau=\inf\{t\ge0:M_t\le\epsilon\}$ is a stopping time.</p>
</blockquote>
<p>I have been able to show that $M_t$ is a martingale but I'm a bit ... | Tsemo Aristide | 280,301 | <p>Consider the serie $\sum_{n\geq 0} (-1)^n/n$, it converges since it is an alternating serie, but $\sum (-1)^{2n}/(2n)=\sum_{n\geq 0}{1\over{2n}}$ does not converge.</p>
|
2,969,195 | <p>I have a cube with rotations <span class="math-container">$\{r, r^2, s, t\}$</span>. <strong>I want to find the cardinality of the conjugacy classes for these elements.</strong> (I know they are 6, 3, 8 and 6 respectively) I couldn't find any formula or anything so I tried to do it by hand for <span class="math-cont... | Burnsba | 114,910 | <p><a href="https://en.wikipedia.org/wiki/Octahedral_symmetry#The_isometries_of_the_cube" rel="nofollow noreferrer">Wikipedia</a> has a good explanation. This is basically the geometric explanation of what your elements are:</p>
<p><span class="math-container">$r$</span>: rotation about an axis from the center of a fa... |
2,148,631 | <p>$a_{n+1} = 1-\frac{1}{a_n + 1}$<br>
$a_1 = \sqrt 2$</p>
<p>I need to prove:<br>
1. $a_n$ is irrational for every $n$.<br>
2. $a_n$ convregres</p>
<p>My ideas: </p>
<ol>
<li>Induction - as I know that $a_1 $ is irrational, I'm assuming $a_n$ is also irrational, and then: $a_{n+1} = 1-\frac{1}{a_n + 1} \implies a... | Fred | 380,717 | <ol>
<li>If $a_n$ is irrational, then suppose that $a_{n+1} = \frac{a_n}{a_n + 1}=\frac{p}{q}$ it follows $a_n= \frac{p}{q-p}$.</li>
</ol>
<p>This gives: if $a_n \notin \mathbb Q$ then $a_{n+1} \notin \mathbb Q$ .</p>
<ol start="2">
<li>You have shown that $ \frac{a_{n+1}}{a_n} = \frac{1}{a_n + 1} \le 1$, hence $a_{n... |
162,147 | <p>I have</p>
<pre><code>J = Table[{x10, y10, x10*y10}, {x10, 0, 1, 0.5}, {y10, 0, 1, 0.5}]
L = Table[{x10, y10, 2.0*x10*y10}, {x10, 0, 1, 0.5}, {y10, 0, 1, 0.5}]
</code></pre>
<p>I want the third elements of J and L to be added and the first and second elements are as they are (as they are the same in both cases) fo... | Akku14 | 34,287 | <p>Another possibility</p>
<pre><code>t3 = Map[(# /. {a_, b_, c_} -> {0, 0, c}) &, t1, {2}] + t2
</code></pre>
|
2,044,322 | <blockquote>
<p>How can you show that :</p>
<p><span class="math-container">$$D_n=\sum_{k=1}^{n } \frac{1}{k}-\int_{1}^{n+1} \frac{1}{x} \ dx $$</span> is increasing and bounded (and hence convergent). I'm having trouble.</p>
</blockquote>
| kccu | 255,727 | <p>Assuming you meant
$$D_n=\sum_{k=1}^n \frac{1}{k}-\int_1^{n+1}\frac{1}{x}\ dx$$
we have
\begin{align*}
D_{n+1}-D_n &= \left(\sum_{k=1}^{n+1} \frac{1}{k}-\int_1^{n+2}\frac{1}{x}\ dx\right)-\left(\sum_{k=1}^n \frac{1}{k}-\int_1^{n+1}\frac{1}{x}\ dx\right)\\
&= \frac{1}{n+1}-\int_1^{n+2}\frac{1}{x} \ dx+\int_1^... |
177,091 | <p>What would $\int\limits_{-\infty}^\infty e^{ikx}dx$ be equal to where $i$ refers to imaginary unit? What steps should I go over to solve this integral? </p>
<p>I saw this in the Fourier transform, and am unsure how to solve this.</p>
| Raymond Manzoni | 21,783 | <p>Draks is right on this (+1) it makes sense as (up to a constant) a representation of the <a href="http://en.wikipedia.org/wiki/Dirac_delta_function#Fourier_kernels" rel="noreferrer">Dirac delta distribution</a> (it is divergent from other points of view!).</p>
<p>More exactly :
$$2\pi \delta(k)=\int\limits_{-\infty... |
3,986,785 | <p>On page 153 of <em>Linear Algebra Done Right</em> the second edition, it says:</p>
<blockquote>
<p>Define a linear map <span class="math-container">$S_1: \text{range}(\sqrt{T^*T} ) \to \text{range}(T)$</span> by:</p>
</blockquote>
<blockquote>
<p><strong>7.43:</strong> <span class="math-container">$S_1 (\sqrt{T^* ... | dperrin | 818,901 | <p>For a function to be well-defined, its output needs to be unambiguous. So if <span class="math-container">$a$</span> = <span class="math-container">$b$</span>, then we need <span class="math-container">$f(a) = f(b)$</span>. Let's look at something that's not well-defined. Let's try to define addition on the rational... |
311,617 | <p>If $S$ is a ring and $R \subset S$ is a subring it's common to write that $S/R$ is an extension of rings. I frequently find myself writing this and read it quite often in textbooks and lecture notes. But whenever I actually think about the notation, I find it to be one of the most confusing conventions in algebra. I... | Pedro | 23,350 | <p>What is true is that $\rm Tor$ commutes with <em>filtered</em> colimits. We already know that $M\otimes -$ commutes with colimits, now take a right $R$-module $M$ and a system $(N_i,\psi_{ji})$ of left $R$-modules over some filtered set $I$. First, we can obtain a short exact sequence of filtered systems </p>
<p>$$... |
4,139,012 | <p>Unfortunately, I don't know much about this topic, that's why I apologize for my language. I have two points <code>P1</code> and <code>P2</code>, now I want "to get from one to the other", but I can only go a distance <code>d</code>. And I want to get as close as possible from <code>P1</code> to <code>P2</... | Siong Thye Goh | 306,553 | <p>We travel along the straight line <span class="math-container">$x=y$</span> where as this line interporlate the two points. We are interested with the case when <span class="math-container">$x>0$</span>.</p>
<p><span class="math-container">$$2x^2=d^2$$</span></p>
<p><span class="math-container">$$x=\frac{d}{\sqr... |
4,139,012 | <p>Unfortunately, I don't know much about this topic, that's why I apologize for my language. I have two points <code>P1</code> and <code>P2</code>, now I want "to get from one to the other", but I can only go a distance <code>d</code>. And I want to get as close as possible from <code>P1</code> to <code>P2</... | Glärbo | 918,887 | <p>Although Siong Thye Goh already provided the <a href="https://math.stackexchange.com/a/4139525/918887">correct answer</a>, I wanted to show the answer explicitly in parametric form.</p>
<p>Let <span class="math-container">$\vec{p}_0$</span> be the starting point, <span class="math-container">$\vec{p}_1$</span> the t... |
527,665 | <p>I need to bound a sum of the form $\sum_{j=1}^m |z+j|^{-n}$ with $\Re z\ge1$ and $n \ge 2$. I am searching for a bound of the form
$$\sum_{j=1}^m |z+j|^{-n} \le C |z|^{-n+1}.$$</p>
<p>It is easy to see that</p>
<p>$$\sum_{j=1}^m |z+j|^{-n} \le \int_{0}^{m+1} |z+x|^{-n}dx \le \int_{0}^{\infty} |z+x|^{-n}dx.$$</p>
... | Community | -1 | <p>We have $(r+x)^2 + s^2 \geq (r+x)^2$. Hence, we have
$$\int_0^{\infty} \dfrac{dx}{((r+x)^2 + s^2)^{n/2}} \leq \int_0^{\infty} \dfrac{dx}{(r+x)^n} = \dfrac{r^{-n+1}}{n-1} < r^{-n+1}$$
We also have $(r+x)^2 + s^2 \geq x^2+s^2$. Hence, we have
\begin{align}
\int_0^{\infty} \dfrac{dx}{((r+x)^2 + s^2)^{n/2}} & \le... |
133,109 | <p>How can I integrate,</p>
<p>$$
\int_n^{+\infty} x \exp\{-ax^2+bx+c\}dx
$$</p>
<p>and what's the result w.r.t the Gaussian function's p.d.f $p(x)$ and c.d.f $\phi(x)$?</p>
<p>Thanks!</p>
| Dilip Sarwate | 15,941 | <p>Hint: Since
$\frac{\mathrm d}{\mathrm dx}\exp(-ax^2+bx+c) = (-2ax+b)\exp(-ax^2+bx+c)$, you can
massage the given integrand to something of the form
$$\frac{-1}{2a}\int (-2ax+b)\exp(-ax^2+bx+c)\ \mathrm dx
+ \int \frac{b}{2a}\exp(-ax^2+bx+c)\ \mathrm dx$$
where the first integral now has a perfect integrand and the... |
1,723,299 | <p>I can work out an heuristic argument for $n=2$: (homeomorphically) turning the disc $D^2$ to something like a funnel (no pipe of course), then gradually contracting the open "mouth" of the funnel to a smaller and smaller hole which eventually vanishes (or "heals", if you imagine the hole as a open wound), we have no... | Community | -1 | <p>There are probably several ways to see this. One way is to realize that the $n$-sphere is homeomorphic to the one point compactification $\mathbb{R}^n\cup\{\infty\}$ of $\mathbb{R}^n$ (you can prove this explicitly using the stereographic projection). Now recall that the open unit $n$-disc $D^n$ in $\mathbb{R}^n$ is... |
32,111 | <p>Let p : Y -> X be an n-sheeted covering map, where X and Y are topological spaces. If X is compact, prove that Y is compact.</p>
<p>I realize that this seems like a very simple problem, but I want to stress the lack of assumptions on X and Y. For example, this is very easy to prove if we can assume that X and Y are... | Georges Elencwajg | 450 | <p>Dear Eric, here is a Bourbaki-style proof.</p>
<p>Recall that a continuous map $f: Y\to X$ is called proper by Bourbaki if, for all spaces $Z$, the map $f\times 1_Z: Y \times Z\to X \times Z$ is closed. For example the trivial finite covering
$X\times \{ 1,\ldots n \}\to X$ is proper.</p>
<p>Now, your $X$ is c... |
619,607 | <p>$\mathbf{(1)}$ Find $y^{\prime}$ of $y=8^{\sqrt x}$ </p>
<p>My try: </p>
<p>$\ln y=\ln(8)^{\sqrt x}$<br>
$\dfrac{1}{y}y^{\prime}=\sqrt{x}\ln8$<br>
I don't know how to proceed with right side. </p>
<p>$\mathbf{(2)}$ Find $y^{\prime}$ of $y=(t+4)(t+6)(t+7).$</p>
<p>This one I have no idea what to do so I don't ha... | QED | 91,884 | <p>$\ln y=\sqrt{x}\ln8$, so $\frac{1}{y}\frac{dy}{dx}=\frac{1}{2\sqrt{x}}\ln8$. Therefore $$y'=\frac{y}{2\sqrt{x}}\ln8=\frac{8^{\sqrt{x}}}{2\sqrt{x}}\ln8$$</p>
<p>For the second question, except for $t=-4,-6,-7$, you may take logarithm of both sides and note that $\frac{1}{dt}\ln(t+a)=\frac{1}{t+a}$, and then carry on... |
3,746,986 | <p>Let <span class="math-container">$\phi:\mathbb (0,\infty) \to [0,\infty)$</span> be a continuous function, and let <span class="math-container">$c \in (0,\infty)$</span> be fixed.</p>
<p>Suppose that "<span class="math-container">$\phi$</span> is convex at <span class="math-container">$c$</span>". i.e. for... | Aryaman Maithani | 427,810 | <p>Define <span class="math-container">$\phi:(-1, \infty) \to [-1, \infty)$</span> as
<span class="math-container">$$\phi(x) = \begin{cases}
\sqrt{1 - (1+x)^2} & x \le 0\\
-x & 0 \le x \le 1 \\
-1 & 1 \le x\end{cases}$$</span></p>
<p>A graph is shown below. (Courtesy of <a href="https://www.desmos.com/calc... |
2,611,450 | <p>I'm having trouble understanding the description of the event in italics (with P = 0.8) in the question below. </p>
<p>"The probability that an American industry will locate in Shanghai is P(S) = 0.7.<br>
The probability that it will locate in Beijing is P(B) = 0.4.<br>
And the <em>probability that it will locate i... | Doug M | 317,162 | <p>What does this relation actually mean?</p>
<p>It really is all about the equivalence of fractions. $\frac {a}{b} = \frac dc \implies ac = bd$</p>
<p>Now, instead of writing these fractions in the more familiar form we show them as and ordered pair.</p>
<p>a,b)</p>
<p>reflexive: </p>
<p>$(a,b)R(a,b) \iff ab = b... |
1,778,601 | <p>Let $G$, $H$ be finite groups. </p>
<p>For a homomorphism $\varphi \colon$ $G \to H$ we learned that for $g \in G$, $ord(\varphi(g))$ must divide $|G|$ and $|H|$. </p>
<p>So then, (in essence), what we did was look at the common factors of $|G|$ and $|H|$ and set $\varphi(x) = \gamma $ where $x$ is a generator of ... | Marko Riedel | 44,883 | <p>Suppose we have $n$ items of $m$ types where $n=km.$ We draw $p$ items
and ask about the expected value of the number of distinct items that
appear.</p>
<p><P>First compute the total count of possible configurations. The
species here is </p>
<p>$$\mathfrak{S}_{=m}(\mathfrak{P}_{\le k}(\mathcal{Z})).$$</p>
... |
4,531,998 | <p>As you increase the value of n, you will generate all pythagorean triples whose first square is even. Is there any visual proof of the following explicit formula and where does it come from or how to derive it?</p>
<p><span class="math-container">$(2n)^2 + (n^2 - 1)^2 = (n^2 + 1)^2$</span></p>
<div class="s-table-co... | Community | -1 | <p>We have the identity <span class="math-container">$a^{2}- b^{2}= \left ( a- b \right )\left ( a+ b \right )\!,$</span> so
<span class="math-container">$$\left ( m^{2}+ 1 \right )^{2}- \left ( m^{2}- 1 \right )^{2}\!=\!2m^{2}\cdot 2= \left ( 2m \right )^{2}$$</span></p>
|
4,531,998 | <p>As you increase the value of n, you will generate all pythagorean triples whose first square is even. Is there any visual proof of the following explicit formula and where does it come from or how to derive it?</p>
<p><span class="math-container">$(2n)^2 + (n^2 - 1)^2 = (n^2 + 1)^2$</span></p>
<div class="s-table-co... | Blue | 409 | <p>Here's a visual proof:</p>
<p><a href="https://i.stack.imgur.com/H7XFV.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/H7XFV.png" alt="enter image description here" /></a></p>
<p>(This space intentionally left blank.)</p>
|
907,729 | <p>I am sorry if the numbers are not formatted, I have searched but found nothing on how. I am trying to multiply $$(2x^3 - x)\left(\sqrt{x} + \frac {2}{x}\right)$$ together and I arrive at a different answer that does not match the one given. The approach I take is to assign the numbers powers then multiply them by th... | Hypergeometricx | 168,053 | <p>Try this:</p>
<p>Put $u^2=x$, (i.e $u=x^{1/2})$:</p>
<p>$$\begin{align}
(2x^3-x)(\sqrt{x}+\frac 2x)&=(2u^6-u^2)(u+\frac2{u^2})\\
&=2u^7+4u^4-u^3-2\\
&=2x^{7/2}+4x^2-x^{3/2}-2\qquad \blacksquare\end{align}$$</p>
|
168,224 | <p>I am doing a project on Hamiltonian group actions on symplectic manifolds, and my supervisor was able to list several good books on Riemannian geometry to start me off, but he didn't know of any single place to learn about Hamiltonian groups. </p>
<p>I have found some books (even available online from the author!) ... | user109527 | 51,275 | <p>The project is now finished, and for anybody else looking to do something similar, I would like to add the following book as an excellent source for an introduction to the material:</p>
<p><a href="http://books.google.co.uk/books?id=5HbJJNYXoN0C&dq=berndt%20symplectic&source=gbs_navlinks_s" rel="nofollow nor... |
3,464,383 | <p>I couldn't find any substantial list of 'strange infinite convergent series' so I wanted to ask the MSE community for some. By <em>strange</em>, I mean infinite series/limits that <strong>converge when you would not expect them to and/or converge to something you would not expect</strong>.</p>
<p>My favorite conver... | Descartes Before the Horse | 592,365 | <p>Another one I like for how simply it is written is as follows:
<span class="math-container">$$\sum_{n=1}^{\infty}z^nH_n = \frac{-\log(1-z)}{1-z}$$</span>
Which holds for <span class="math-container">$|z|<1$</span>, <span class="math-container">$H_n$</span> being the <span class="math-container">$n$</span>-th harm... |
4,293,960 | <p>I assume discrete time. Suppose I start with some amount of money <span class="math-container">$P_{0}$</span>. Then, using a simple rate of interest <span class="math-container">$r$</span> for a given period of time, I would have</p>
<p><span class="math-container">$$P_{t}= P_{0} (1+r)^{t}$$</span></p>
<p>Suppose <s... | user58697 | 58,697 | <blockquote>
<p>Now, suppose I want the payments to happen more frequently, say <span class="math-container">$n$</span>
times a year, but I want <span class="math-container">$t$</span> to still represent years. Then I can model the amount I get paid as <span class="math-container">$P_{t}= P_{0} (1+r)^{nt}$</span></p>
<... |
4,293,960 | <p>I assume discrete time. Suppose I start with some amount of money <span class="math-container">$P_{0}$</span>. Then, using a simple rate of interest <span class="math-container">$r$</span> for a given period of time, I would have</p>
<p><span class="math-container">$$P_{t}= P_{0} (1+r)^{t}$$</span></p>
<p>Suppose <s... | Misha Lavrov | 383,078 | <p>In principle you could put any function <span class="math-container">$f(n)$</span> anywhere you like. The bank is allowed to pay interest according to any rule that it's willing to explain to confused clients. But using <span class="math-container">$\frac rn$</span> isn't arbitrary: it comes out of some generalizati... |
294,972 | <p>I am seeking a particular integral of the differential equation:</p>
<p>$u^{(4)}(t) - 5u''(t) + 4u(t) - t^3 = 0$</p>
<p>I simply interested in the technique, not just an answer (Mathematica suffices for that). How would one apply undetermined coefficients in this case?</p>
| vonbrand | 43,946 | <p>The standard <em>general</em> technique is <a href="http://www.cliffsnotes.com/study_guide/Variation-of-Parameters.topicArticleId-19736,articleId-19722.html" rel="nofollow">variation of parameters</a>. In this case, as the ODE is linear with constant coefficients and the RHS is a polynomial, just try a $3 + 4 = 7$ d... |
3,848,676 | <p>Let <span class="math-container">$n$</span> and <span class="math-container">$k$</span> be nonnegative integers. The binomial coefficient <span class="math-container">$\binom{n}{k}$</span> is equal to the number of ways to choose <span class="math-container">$k$</span> items from a set with <span class="math-contain... | Vinyl_cape_jawa | 151,763 | <p>HINT:</p>
<p>The Binomial Theorem says:
<span class="math-container">$$
(a+b)^n=\sum_{k=0}^{n}\binom{n}{k} a^{n-k}b^k,
$$</span>
where the left hand side is
<span class="math-container">$$
\underbrace{(a+b)\cdot(a+b)\cdot(a+b)\cdot\ldots\cdot(a+b)}_{n \ times}.
$$</span>
If you would expand this product you would p... |
2,010,768 | <p>I have the question </p>
<p>Solve the simultaneous equations </p>
<p>$$\begin{cases}
3^{x-1} = 9^{2y} \\
8^{x-2} = 4^{1+y}
\end{cases}$$</p>
<p><a href="https://i.stack.imgur.com/xN9Ny.jpg" rel="nofollow noreferrer">source image</a></p>
<p>I know that $x-1=4y$ and $3X-6=2+2y$ .</p>
<p>However when I checked th... | Peter | 82,961 | <p>Look at the second equation : $3x-6=2+2y$. If we subtract $2$ and multiply with $2$, we get $6x-16=4y$</p>
|
818,850 | <p>I want to generate numbers $1$ to $10$ with uniform probability distribution. So I write the numbers $1$ to $10$ in the natural order. I keep writing the next $10$ number block by permuting the first $10$ numbers. Will this long string can be considered random with uniform distribution? Is there a problem?</p>
| Anon | 155,030 | <p>Here's one way to visualize an infinite series adding up to a finite value:</p>
<p>A flea is 1 metre away from a wall. The flea jumps half way to the wall (1/2). Then it jumps half way again (1/4). And again (1/8). It keeps going forever.</p>
<p>After a large number of jumps, it gets really close to the wall. ... |
4,004,457 | <p>I searched but couldn't find the proof.</p>
<p><strong>Isosceles hyperbola equation</strong>:
<span class="math-container">$${H:x^{2}-y^{2} = a^{2}}$$</span></p>
<p>And let's take <strong>any point <span class="math-container">$P(x, y)$</span></strong> on this hyperbola. Now, the product of the distances of this poi... | Z Ahmed | 671,540 | <p>The missing no is 6. Sum of numbers in each horizontal layer is 36.</p>
|
4,004,457 | <p>I searched but couldn't find the proof.</p>
<p><strong>Isosceles hyperbola equation</strong>:
<span class="math-container">$${H:x^{2}-y^{2} = a^{2}}$$</span></p>
<p>And let's take <strong>any point <span class="math-container">$P(x, y)$</span></strong> on this hyperbola. Now, the product of the distances of this poi... | Dietrich Burde | 83,966 | <p>The number is <span class="math-container">$n=66$</span>. The sum of all alternating sums of the rows should equal zero. So we need
<span class="math-container">$$
36+(30-n)-8+8=0,
$$</span>
which says that <span class="math-container">$n=66$</span>.
For example, the last row has alternating sum <span class="math-c... |
2,857,508 | <p>I'm trying to prove this: $$0.99^n \le 1/2,\text{ for }n=100$$</p>
<p>I tried Bernoulli's inequality $(1-0.01)^n \geq 1-n\cdot0.01$ and it gave me LHS $\geq0$. </p>
<p>I also tried to do this: $((1-0.01)^n)^n \leq(1-1/2)^n$ and it gave me LHS $\geq-99$ and RHS $\geq49$. </p>
<p>Now I'm stuck.</p>
| Batominovski | 72,152 | <p>Note that
$$\left(1+\frac{1}{99}\right)^n \geq 1 +\frac{n}{99}$$
for every $n\geq 1$. In particular,
$$\frac{1}{0.99^{100}}=\left(1+\frac{1}{99}\right)^{100}\geq 1+\frac{100}{99}>2\,.$$</p>
|
1,056,397 | <p>I saw the following problem:
$$\lim_{x\to \infty} \sqrt{9x^2+x}-3x$$
My first thought was to say that the $x$ term is overpowered when $x$ becomes large enough, so the square root becomes just $\sqrt{9x^2} = 3x$, and the value of the limit is zero.</p>
<p>However, the solution given is $1\over6$, and starts with:
$... | Vincenzo Oliva | 170,489 | <p>Because the original limit is an indeterminate form, where that $x$ moves you a bit further from $0$. At any rate, formally you're not allowed to simply drop a term. Initially, as you can see, the <a href="http://demonstrations.wolfram.com/FundamentalLawOfFractions/" rel="nofollow">fundamental law of fractions</a> ... |
248,167 | <p>I want to know why, when I look at the Julia sets of the quadratic family, I see only a finite number of repeating patterns, rather than a countable infinity of them.</p>
<p>My question is specifically about the interaction of these three theorems:</p>
<p><strong>Theorem 1</strong>: Let $z_0\in\mathbb{C}$ be an re... | Lasse Rempe | 3,651 | <p>As @GNiklasch points out, you seem to be zooming into two places which are both preimages of the same repelling periodic point. So the images of the Julia set are locally related by a conformal map, and hence indeed asymptotically the same.</p>
<p>If you zoom in at different repelling periodic points, then generall... |
3,366,981 | <p>I have been looking for a concrete answer on this matter since leaving secondary school and thought now to ask online since YouTube and Wikipedia seem too convoluted. </p>
<p>I remember watching <a href="https://www.youtube.com/watch?v=Rzac520CESc" rel="nofollow noreferrer">this</a> video on the proof behind the nC... | Batominovski | 72,152 | <p>Without doing the integration, notice first that <span class="math-container">$I_x+I_y=I_z$</span>. Also, <span class="math-container">$I_x=I_y$</span> due to symmetry, so <span class="math-container">$I_x=I_y=\frac{1}{2}I_z$</span>. But for <span class="math-container">$I_z$</span>, every part of the ring is at t... |
190,497 | <p>Q: Is there a reference for a detailed proof of <a href="http://en.wikipedia.org/wiki/Explicit_formula" rel="nofollow">Riemann's explicit formula</a> ?</p>
<p>I am reading the <a href="http://www.claymath.org/millennium/Riemann_Hypothesis/1859_manuscript/EZeta.pdf" rel="nofollow">TeXified version</a> of Riemann's m... | Gerry Myerson | 8,269 | <p>Answering the question in the title, not the body, Riemann's explicit formula is stated on page 244 of Stopple, A Primer of Analytic Number Theory, and discussed over the next several pages. By the way, it's considered that Riemann only gave a "heuristic proof," the first rigorous proof being given by von Mangoldt i... |
2,982,723 | <p>I need help in this exercise. What I need to prove that the function <span class="math-container">$f$</span> given is not continuous in the point <span class="math-container">$(0,0)$</span></p>
<p><span class="math-container">$$ f(x,y) = \begin {cases} \frac {x^3\times y} {x^6+y^2} & (x,y) \not = 0\\ 0 & (... | user | 505,767 | <p><strong>HINT</strong></p>
<p>We have that by <span class="math-container">$x=u$</span> and <span class="math-container">$y=v^3$</span></p>
<p><span class="math-container">$$\frac {x^3 y} {x^6+y^2}=\frac {u^3 v^3} {u^6+v^6}$$</span></p>
<p>which is a simpler case to handle.</p>
|
2,251,112 | <p>A small college has 1095 students. What is the approximate probability that more than five students were born on Christmas day? Assume that the birthrates are constant throughout the year and that each year has 365 days.</p>
<p>I tried doing<br>
$X \sim Pn(3) $ and calculating $P(X\gt5)$. My calculation turned out ... | WhiteDeer | 502,721 | <p>N should be 5.5 not 4.5</p>
<p>P(X>5) = P(N >5.5) // continuity correction</p>
<p>N~(μ,σ^2)</p>
<p>N~(3,2.99) <--- σ^2 should not be (1092/365)</p>
<p>μ = n<em>p(amount of students)</em>(probability of person born on Christmas Day)</p>
<p>=1095 *(1/365)</p>
<p>=3</p>
<p>σ = sqrt(n*p(1-p))</p>
<p>= sqrt(10... |
76,747 | <pre><code>Plot[D[Abs[x], x], {x, -10, 10}, Exclusions -> {0}]
</code></pre>
<p>gives out several lines of error messages and an empty plot.</p>
<pre><code>Plot[Derivative[1][Abs[#1] & ][x], {x, -10, 10}, Exclusions -> {0}]
</code></pre>
<p>just gives out an empty plot.</p>
<p>How do I plot $\left(\left|x... | Basheer Algohi | 13,548 | <p>you can also use <code>UnitStep</code>:</p>
<pre><code>f[a_] := (x - a) (UnitStep[x - a] - UnitStep[-(x - a)])
Plot[Evaluate@D[f[5], x], {x, -10, 10}, Exclusions -> 0]
</code></pre>
<p><img src="https://i.stack.imgur.com/0aNZJ.jpg" alt="enter image description here"></p>
|
3,851,399 | <p>I am attempting to prove the following statement</p>
<blockquote>
<p>Prove that if n is any integer then 4 either divides n^2 or n^2 − 1</p>
</blockquote>
<p>I have started with the case of n = 2k</p>
<pre><code>Consider the case n = 2k
n = 2k
n^2 = 4k^2
⇒ n = 4k
∴ 4 divides n^2 as there is some integer k in w... | Servaes | 30,382 | <p>Your argument for the first case is fine, except for the very last few words:</p>
<p><code>∴ 4 divides n^2 as there is some integer k in which n = k </code></p>
<p>You have already declared before that <code>n=2k</code>. To now refer to <code>k</code> again and claim that <code>n=k</code> is confusing at best, and i... |
1,226,176 | <p>The following serie is obviously convergent, but I can not figure out how to calculate its sum :</p>
<p>$$\sum \frac{6^n}{(3^{n+1}-2^{n+1})(3^n - 2^n)}$$</p>
| imranfat | 64,546 | <p>Allright. Time to get some excel in here. After playing with it, it shows that the limit goes to 4. Now let's try to prove that.</p>
<p>Let's set up partial fractions, as first suggested by Steven.</p>
<p>I did: $\frac{A}{3^{n+1}-2^{n+1}}+\frac{B}{3^n-2^n}$ Adding these terms together by making a common denominato... |
1,879,346 | <p>Ok, so I have this question from my math book.</p>
<blockquote>
<p>The vertices A,B,C of a triangle are (3,5),(2,6) and (-4,-2)
respectively. Find the coordinates of the circum-centre and also the
radius of the circum-circle of the triangle.</p>
</blockquote>
<p>How can we solve this? Can we use the distance... | Emilio Novati | 187,568 | <p>Hint:</p>
<p>You want a circumference that passe thorough the three given points. </p>
<p>The general equation of a circumference is $x^2+y^2+ax+by+c=0$.</p>
<p>Substitute the coordinates of the three points and you have a linear system in the three unknowns $a,b,c$. </p>
<p>Solve this system and you have the eq... |
1,789,414 | <blockquote>
<p><strong>Question :</strong> Prove that the equation $x + \cos(x) + e^{x} = 0$ has
<em>exactly</em> one root</p>
</blockquote>
<hr>
<p><em>This is what I thought of doing:</em></p>
<p>$$\text{Let} \ \ \ f(x) = x + \cos(x) + e^{x}$$
By using the Intermediate Value Theorem on the open interval $(-\... | openspace | 243,510 | <p>Consider the first derivate of it:
$f'(x)=1-\sin(x) + e^{x} \ge e^{x}$, for $x \in R$.
So function is grow up , now you should use the Rolle's theorem</p>
|
419,730 | <p>$EX = \int xf(x)dx$, where $f(x)$ is the density function of some X variable. This is pretty understandable, but what if I had, for example an $Y = -X + 1$ function and I had to calculate $EY$? How should I do this?</p>
| Adriano | 76,987 | <p>By the linearity of expectation, we have:
$$
E(Y) = E(-X + 1) = -E(X) + 1
$$</p>
<p>Alternatively, you could use the definition directly. Let $Y=g(X)=-X+1$. Then:
$$ \begin{align*}
E(Y)&=E(g(X))\\
&=\int_{-\infty}^\infty g(x)f(x)dx\\
&=\int_{-\infty}^\infty(-x+1)f(x)dx\\
&=-\int_{-\infty}^\infty xf(... |
382,295 | <p>Please help. I haven't found any text on how to prove by induction this sort of problem: </p>
<p>$$
\lim_{n\to +\infty}1 + \frac{1}{4} + \frac{1}{4^2} + \cdots+ \frac{1}{4^n} = \frac{4}{3}
$$</p>
<p>I can't quite get how one can prove such. I can prove basic divisible "inductions" but not this. Thanks.</p>
| TonyK | 1,508 | <p>If you want to use proof by induction, you have to prove the stronger statement that</p>
<p>$$
1 + \frac{1}{4} + \frac{1}{4^2} + \cdots+ \frac{1}{4^n} = \frac{4}{3} - \frac{1}{3}\frac{1}{4^n}
$$</p>
|
2,325,650 | <p>This question is based on the fourth question in the 2003 edition of the <a href="https://www.vwo.be/vwo/files/finale03.pdf" rel="nofollow noreferrer">Flemish Mathematics Olympiad</a>.</p>
<blockquote>
<p>Consider a grid of points with integer coordinates. If one chooses the number $R$ appropriately, the circle w... | jgsmath | 455,126 | <p>Following arguments are not mathematically rigorous, but I think it will explain the main idea.</p>
<p>The solution hinges on the fact that there are infinitely many values of $\phi \in [0, 2\pi)$ such that $\cos{\phi}$ and $\sin{\phi}$ are both rational. (This in turn depends on the fact that there are infinitely... |
3,043,319 | <p>I just begin with the 3- dimension function. I dont really understand how to begin with this problem.
Function <span class="math-container">$$f(x,y)= (x^2+4y^2-5)(x-1)$$</span>
Find where <span class="math-container">$$f(x,y)=0; f(x,y)>0; f(x,y)<0.$$</span></p>
<p>To find where <span class="math-container">$... | Community | -1 | <p>We only need to show three properties: </p>
<p><strong>Reflexiveness:</strong></p>
<p>Does there exist for ANY set <span class="math-container">$A$</span> a bijective function between the set and itself?
If so, we can say that <span class="math-container">$A \sim A$</span>.</p>
<p>Hint, what if there were a func... |
7,725 | <p>Basically the textbooks in my country are awful, so I searched on the web for a precalculus book and found this one: <a href="http://www.stitz-zeager.com/szprecalculus07042013.pdf">http://www.stitz-zeager.com/szprecalculus07042013.pdf</a></p>
<p>However, it does not cover convergence,limits etc. and those topics we... | schremmer | 4,815 | <p>Strictly speaking, the following might be slightly off topic, but <em>I</em> think that it is directly relevant. </p>
<p>From the preface of Calculus with Analytic Geometry by John H Staib [1966],</p>
<blockquote>
<p>The development of the book is committed to the following thesis: The
theory of limits is most... |
616,454 | <p>How can I calculate this integral?</p>
<p>$$ \int {\exp(2x)}{\sin(3x)}\, \mathrm{d}x$$ </p>
<p>I tried using integration by parts, but it doesn't lead me any improvement. So I made an attempt through the replacement $$ \cos(3x) = t$$ and it becomes $$\frac{1}{-3}\int \exp\left(2\left(\dfrac{\arccos(t)}{3}\right)... | ncmathsadist | 4,154 | <p>You need to integrate by parts twice, either differentiating the exponential each time or differentiating the trig function each time. You will be able to solve for the integral.</p>
|
2,610,560 | <blockquote>
<p>Given that $x, y, z$ are positive integers and that $$3x=4y=7z$$ find
the minimum value of $x+y+z$. The options are:</p>
<p>A) 33</p>
<p>B) 40</p>
<p>C) 49</p>
<p>D) 61</p>
<p>E) 84</p>
</blockquote>
<p>My attempt:</p>
<p>$y=\frac{3}{4}x, z=\frac{3}{7}x$. </p>
<p>Substit... | Mark Neuhaus | 21,706 | <p>This is a classical problem from <em>integer programming</em>. There are exact algorithmic solutions available (although they are slow, as it is known that this a NP-hard problem). You can look for "branch and bound" or "branch and cut" or <em>eventually</em> total unimodularity. It basically works by a cleaver ext... |
2,314,369 | <p>My objective is to represent a unit square using a single matrix.<br>The four corners of the unit square are represented by column vectors as follows,</p>
<p>$\left[\begin{matrix} 0 \\ 0 \end{matrix}\right],\left[\begin{matrix} 1 \\ 0 \end{matrix}\right],\left[\begin{matrix} 1 \\ 1 \end{matrix}\right],\left[\begin{... | Jean Marie | 305,862 | <p>You should transpose this matrix:</p>
<p>$$C=\left[\begin{matrix} 0&1&1&0 \\ 0&0&1&1 \end{matrix}\right]$$</p>
<p>With this presentation, any linear operation $L$ on $C$ will be $C'$ given by $L \times C=:C'$; for example, </p>
<p>1) The orthogonal reflection with respect to the $x$ axis (... |
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