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 |
|---|---|---|---|---|
4,614,853 | <p>I'm trying to prove that the BinPacking problem is NP hard granted the partition problem <em>is NP hard</em>. If I have E a set of positive integers, can I split it into two subsets such that the sums of the integers in both subsets are equal?</p>
<p>The polynomial reduction I found would be the following:</p>
<ul>
... | Paul DUBOIS | 1,138,237 | <p>It is not wrong, you effectively reduce only some of the BinPacking problem instances.
You will only have an <em>explicit</em> solution for BinPacking with 2 bags, this is correct also.</p>
|
78,311 | <p>Let $\mu$ be standard Gaussian measure on $\mathbb{R}^n$, i.e. $d\mu = (2\pi)^{-n/2} e^{-|x|^2/2} dx$, and define the Gaussian Sobolev space $H^1(\mu)$ to be the completion of $C_c^\infty(\mathbb{R}^n)$ under the inner product
$$\langle f,g \rangle_{H^1(\mu)} := \int f g\, d\mu + \int \nabla f \cdot \nabla g\, d\mu.... | Nate Eldredge | 822 | <p>Byron's paper, which he linked in his (accepted) answer, has a proof in a more general setting (where the Gaussian measure can be replaced by any measure with exponentially decaying tails). Here is a specialization of it to the Gaussian case, which I wrote up to include in some lecture notes. I guess I was on the ... |
78,311 | <p>Let $\mu$ be standard Gaussian measure on $\mathbb{R}^n$, i.e. $d\mu = (2\pi)^{-n/2} e^{-|x|^2/2} dx$, and define the Gaussian Sobolev space $H^1(\mu)$ to be the completion of $C_c^\infty(\mathbb{R}^n)$ under the inner product
$$\langle f,g \rangle_{H^1(\mu)} := \int f g\, d\mu + \int \nabla f \cdot \nabla g\, d\mu.... | JT_NL | 1,120 | <p>I think I have a different proof. Let $\gamma$ be the <em>Gaussian measure</em>, that is, $\gamma$ is given by the <em>Radon-Nikodym density</em>,
$$\mathrm{d}\gamma(x) = \frac{\mathrm{e}^{-x^2}}{\sqrt{\pi}} \mathrm{d}x.$$
Also, consider the <em>Ornstein-Uhlenbeck operator</em> given as,
$$L := -\frac12 \Delta + x \... |
1,600,054 | <p>The graph of $y=x^x$ looks like this:</p>
<p><a href="https://i.stack.imgur.com/JdbSv.gif" rel="noreferrer"><img src="https://i.stack.imgur.com/JdbSv.gif" alt="Graph of y=x^x."></a></p>
<p>As we can see, the graph has a minimum value at a turning point. According to WolframAlpha, this point is at $x=1/e$.</p>
<p>... | sirfoga | 83,083 | <p><strong>Hint</strong>: actually you are looking for a local/global minimum .. so look at the derivative of the function $f(x) = x^x$
$$f'(x) = x^x (\log (x)+1)$$
which equals $0 \iff x = \frac{1}{e}$ </p>
|
1,600,054 | <p>The graph of $y=x^x$ looks like this:</p>
<p><a href="https://i.stack.imgur.com/JdbSv.gif" rel="noreferrer"><img src="https://i.stack.imgur.com/JdbSv.gif" alt="Graph of y=x^x."></a></p>
<p>As we can see, the graph has a minimum value at a turning point. According to WolframAlpha, this point is at $x=1/e$.</p>
<p>... | Brenton | 226,184 | <p>Hint: Note that $x^x = e^{x\log x}$.</p>
<p>So minimizing $x^x$ is the same as minimizing $x\log x$</p>
|
1,988,563 | <blockquote>
<p>Use the formal defintion to prove the given limit:
$$\lim_{x\to\frac13^+}\sqrt{\frac{3x-1}2}=0$$</p>
</blockquote>
<p>Not sure how to deal with $\sqrt\cdot$. Appreciate a hint.</p>
| Doug M | 317,176 | <p>$\forall \epsilon>0,\exists \delta>0\text{ such that } 0<(x-\frac 13)<\delta \implies |\sqrt{\frac{3x-1}{2}}|<\epsilon$</p>
<p>$\sqrt{\frac{3x-1}{2}} < \sqrt{\frac 32} \sqrt \delta$</p>
<p>$\delta \le \frac 23 \epsilon^2\implies|\sqrt{\frac{3x-1}{2}}|<\epsilon$</p>
|
1,988,563 | <blockquote>
<p>Use the formal defintion to prove the given limit:
$$\lim_{x\to\frac13^+}\sqrt{\frac{3x-1}2}=0$$</p>
</blockquote>
<p>Not sure how to deal with $\sqrt\cdot$. Appreciate a hint.</p>
| Masacroso | 173,262 | <p>By the axioms of order of a field we know that multiplying both sides of an inequality by a positive quantity the inequality doesnt change, then</p>
<p>$$\left|\sqrt{\frac{3x-1}2}\right|<\epsilon\iff
\left|\frac{3x-1}2\right|<\epsilon\cdot \left|\sqrt{\frac{3x-1}2}\right|<\epsilon^2$$</p>
<p>where the la... |
3,422,830 | <blockquote>
<p>In the polynomial
<span class="math-container">$$
(x-1)(x^2-2)(x^3-3) \ldots (x^{11}-11)
$$</span>
what is the coefficient of <span class="math-container">$x^{60}$</span>? </p>
</blockquote>
<p>I've been trying to solve this question since a long time but I couldn't. I don't know whether opening ... | amir bahadory | 204,172 | <p>Hint : <span class="math-container">$1+2+3 +...+11= \frac {11×12}{2} =66 $</span> so we must find how we can construct number <span class="math-container">$6=6+0=5+1=4+2=3+3=1+2+3$</span> and note that <span class="math-container">$3+3$</span> impossible. </p>
|
3,422,830 | <blockquote>
<p>In the polynomial
<span class="math-container">$$
(x-1)(x^2-2)(x^3-3) \ldots (x^{11}-11)
$$</span>
what is the coefficient of <span class="math-container">$x^{60}$</span>? </p>
</blockquote>
<p>I've been trying to solve this question since a long time but I couldn't. I don't know whether opening ... | A.J. | 654,406 | <p>Expanding the brackets is definitely not the way to go, but thinking about what would happen if you did is helpful. Every term in the expanded polynomial will come from multiplying one term from each bracket (e.g. the highest degree term will come from multiplying <span class="math-container">$\,x \cdot x^2 \cdot x^... |
3,380,998 | <p>Is it possible to express the cube root of "i" without using "i" itself?</p>
<p>If this is possible can you show me how to arrive at it?</p>
<p>thanks</p>
| Henno Brandsma | 4,280 | <p>No. <span class="math-container">$i$</span> has 3 cube roots in the complex numbers and none of them can of course be real (i.e. have no imaginary part; a real number has a real third power, not <span class="math-container">$i$</span>). They are </p>
<p><span class="math-container">$$e^{i\frac{\pi}{6}}, e^{i\frac{5... |
104,297 | <p>How would I go about solving</p>
<p>$(1+i)^n = (1+\sqrt{3}i)^m$ for integer $m$ and $n$?</p>
<p>I have tried </p>
<pre><code>Solve[(1+I)^n == (1+Sqrt[3] I)^m && n ∈ Integers && m ∈ Integers, {n, m}]
</code></pre>
<p>but this does not give the answer in the 'correct' form.</p>
| bbgodfrey | 1,063 | <p>As stated in the question and also the comment above by <a href="https://mathematica.stackexchange.com/users/12/szabolcs">Szabolcs</a>, Mathematica does not seem to be able to solve the equation directly. For instance, neither <code>Solve</code> nor <code>Reduce</code> produces the desired result. However, as I su... |
2,239,192 | <p>Let $P_n$ be the polynomials of degree no more than n with basis $Z_n=(1, x, x^2,\dotsc,x^n)$. The derivative transformation $D$ goes from $P_n$ to $P_{n-1}$. Write out the matrix for $D$ from $(P_4, Z_4)$ to $(P_3, Z_3)$.</p>
<p>I haven't done a problem similar to this so I'm not sure how to go about doing this. ... | Brian Fitzpatrick | 56,960 | <p>We have bases
\begin{align*} \alpha &= \{1,x,x^2,x^3,x^4\} & \beta &= \{1,x,x^2,x^3\}
\end{align*} for $P_4$ and $P_3$ respectively.</p>
<p>Our map $D:P_4\to P_3$ is given by $D(f)=f^\prime$. To compute
$[D]_\alpha^\beta$, we must evaluate $D$ on each basis element in $\alpha$ and
write the output in te... |
310,930 | <p>Let $U$ be the subspace of $\mathbb{R}^3$ spanned by $\{(1,1,0), (0,1,1)\}$. Find a subspace $W$ of $\Bbb R^3$ such that $\mathbb{R}^3 = U \oplus W$.</p>
<p>As I am having an examination tomorrow, it would be really helpful if one could explain the methodology for doing this problem. I am mostly interested in the m... | Jim | 56,747 | <p>In general, if you have a subspace $U \subseteq V$ and you want a complement $W$ so that $U \oplus W = V$. Then first find a basis $\{u_1, \ldots, u_n\}$ of $U$. Then extend this to a basis $\{u_1, \ldots, u_n, w_1, \ldots, w_m\}$ of $V$. The complement is the subspace spanned by those additional vectors,
$$W = \... |
3,226,320 | <p>I have some data points that need to be fit to the curve defined by</p>
<p><span class="math-container">$$y(x)=\frac{k}{(x+a)^2} - b$$</span></p>
<p>I have considered that it can be done by the least squares method. However, the analytical solution gives me a negative <span class="math-container">$a$</span>, so it... | Claude Leibovici | 82,404 | <p>In any manner, your model is nonlinear with respect to parameters. So, why not to rewrite it as
<span class="math-container">$$y(x)=\frac{k}{(x+\alpha^2)2} - b$$</span></p>
|
1,408,467 | <p>I've found a few papers that deal with removing redundant inequality constraints for linear programs, but I'm only trying to find the non-redundant constraints that define a feasible region (i.e. I have no objective function), given a set of possibly redundant inequality constraints.</p>
<p>For instance, if I have:... | tomi | 215,986 | <p>Suppose you have four non-redundant constraints. These define the feasible region (some sort of quadrilateral).</p>
<p>The fifth constraint is redundant if it does not intersect the feasible region - if it is non-redundant then adding it to the problem will trim off some part opf the feasible region.</p>
<p>For ea... |
89,000 | <p>Let $f:I \rightarrow \mathbb{R}$, where $I\subset \mathbb{R}$ is an interval, be midconvex, that is
$$f\left(\frac{x+y}{2}\right) \leq \frac{f(x)+f(y)}{2}$$ for all $x,y \in I$.
Assume that for some $x_0, y_0 \in \mathbb{R}$ such that $x_0 < y_0$ holds equality
$$f\left(\frac{x_0+y_0}{2}\right)= \frac{f(x_0)+f(... | Martin Sleziak | 8,297 | <p>If I understand your new question correctly (after the addition in
your question and some clarification in the comments) you want to
know, whether the following is true:</p>
<blockquote>
<p>Let $f$ be a midpoint convex (a.k.a. Jensen convex) function on
$[x_0,y_0]$, i.e. for any $x,y\in[x_0,y_0]$
$$ f\left(\f... |
3,571,047 | <p>Here's what I have so far:</p>
<p><span class="math-container">$$\frac{\partial f}{\partial y}|_{(a,b)} = \lim\limits_{t\to 0} \frac{\sin(a^2 + b^2 + 2tb + t^2) - \sin(a^2 + b^2)}{t} = \lim\limits_{t\to 0} \frac{\sin(a^2 + b^2)[\cos(2tb + t^2) - 1] + \cos(a^2 + b^2)\sin(2tb + t^2)}{t}$$</span>
I can see that side l... | Boka Peer | 304,326 | <p>You said <span class="math-container">$t = 1 + \sqrt{x}$</span> gives <span class="math-container">$dt = x + 2x^{3/2}/3 + C,$</span> which is not correct. It seems instead of taking derivative, you took integral. I am giving you some hinits. </p>
<p>Suppose <span class="math-container">$t = 1 + \sqrt{x}$</span>. Th... |
81,811 | <p>I heard this example was given in Whitehead's paper A CERTAIN OPEN MANIFOLD WHOSE GROUP IS UNITY.( <a href="http://qjmath.oxfordjournals.org/content/os-6/1/268.full.pdf" rel="nofollow">http://qjmath.oxfordjournals.org/content/os-6/1/268.full.pdf</a> ) But I was confused by his term. Thus I'm looking for an explanati... | Autumn Kent | 1,335 | <p>The manifold is the <a href="http://en.wikipedia.org/wiki/Whitehead_manifold" rel="nofollow">Whitehead manifold</a>.</p>
|
1,364,430 | <p><strong>Problem</strong></p>
<p>How many of the numbers in $A=\{1!,2!,...,2015!\}$ are square numbers?</p>
<p><strong>My thoughts</strong></p>
<p>I have no idea where to begin. I see no immediate connection between a factorial and a possible square. Much less for such ridiculously high numbers as $2015!$.</p>
<p... | ajotatxe | 132,456 | <p>Only $1!$. For $n>1$, let $p$ be the greatest prime with $p\le n$. <a href="https://en.wikipedia.org/wiki/Bertrand%27s_postulate">Between $p$ and $2p$ there is another prime</a>, so $2p>n$. Therefore, $p$ occurs only once in the factorization of $n!$ and hence, $n!$ is not a square.</p>
|
117,619 | <p>I need to evaluate the following real convergent improper integral using residue theory (vital that i use residue theory so other methods are not needed here)
I also need to use the following contour (specifically a keyhole contour to exclude the branch cut):</p>
<p><a href="https://i.stack.imgur.com/4wwwj.png" rel... | Robert Israel | 8,508 | <p>Consider the integral of $\sqrt{z}/(z^3+1)$ around the given contour, using a branch of $\sqrt{z}$ with branch cut on the positive real axis. This can be evaluated using residues.
Note that (in the appropriate limit) the integrals over $L_1$ and $L_2$ both approach $\int_0^\infty \frac{\sqrt{x}}{x^3+1}\ dx$ (for $L... |
2,873,520 | <p>I want to find out how interference of two sine waves can affect the output-phase of the interfered wave. </p>
<p>Consider two waves,</p>
<p>$$ E_1 = \sin(x) \\
E_2 = 2 \sin{(x + \delta)}
$$</p>
<p>First off, I don't know how to prove it, but I can see visually (plotting numerically) that the sum of these waves... | Chickenmancer | 385,781 | <p>There is a rich study of so-called "differential algebra."</p>
<p><a href="https://en.wikipedia.org/wiki/Differential_algebra" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Differential_algebra</a></p>
<p>However, what you're realizing is the connection between linear algebra and differential equations. ... |
743,988 | <p>If we formally exponentiate the derivative operator $\frac{d}{dx}$ on $\mathbb{R}$, we get</p>
<p>$$e^\frac{d}{dx} = I+\frac{d}{dx}+\frac{1}{2!}\frac{d^2}{dx^2}+\frac{1}{3!}\frac{d^3}{dx^3}+ \cdots$$</p>
<p>Applying this operator to a real analytic function, we have</p>
<p>$$\begin{align*}e^\frac{d}{dx} f(x) &... | Stephen Montgomery-Smith | 22,016 | <p>So you have suggested <span class="math-container">$e^{t \frac d{dx}} f(x) = f(x+t)$</span>. This is to be expected, because you would formally expect (i) <span class="math-container">$e^{0 \frac d{dx}}$</span> to be the identity operator, (ii) <span class="math-container">$\frac d{dt} [e^{t \frac d{dx}} f(x)] \big... |
1,441,349 | <p>Given is that $a∈ℤ_n^*$ and $d|ord(a)$.</p>
<p>I need to show that $ord(a^d)= ord(a)/d$.</p>
<p>I started with the following:</p>
<p>$ord(a^d) = e$, such that $(a^d)^e\equiv 1\pmod n$</p>
<p>$ord(a)/d =f/d$ where $ord(a)=f$, such that $a^f\equiv 1\pmod n$</p>
<p>Now I want to prove that $e=f/d$. </p>
<p>I have... | Mauro ALLEGRANZA | 108,274 | <p>We say that :</p>
<blockquote>
<p>$\Gamma$ <em>tautologically implies</em> (or <em>logically implies</em> or <em>semantically entails</em>) $\varphi$ (written $\Gamma \vDash \varphi$) iff every truth assignment for the sentence symbols in $\Gamma$ and $\varphi$ that satisfies every member of $\Gamma$ also satisfi... |
366,654 | <p>Find all values of real number p or which the series converges:</p>
<p>$$\sum \limits_{k=2}^{\infty} \frac{1}{\sqrt{k} (k^{p} - 1)}$$ </p>
<p>I tried using the root test and the ratio test, but I got stuck on both. </p>
| André Nicolas | 6,312 | <p>Things will look nicer if we let $y=w^2$. Then $\frac{dy}{dx}=2w\frac{dw}{dx}$ and we end up with
$$2w\frac{dw}{dx}+\frac{w^2}{x-2}=5(x-2)w.$$
There is the solution $w=0$. For others, cancel. We get a nice linear equation. The $x-2$ is slightly annoying, at least for typing, so let $t=x-2$. We have arrived at
$$2\f... |
3,460,426 | <p>I tried to take the <span class="math-container">$Log$</span> of <span class="math-container">$\prod _{m\ge 1} \frac{1+\exp(i2\pi \cdot3^{-m})}{2} = \prod _{m\ge 1} Z_m$</span>, which gives </p>
<p><span class="math-container">$$Log \prod_{m\ge 1} Z_m = \sum_{m \ge 1} Log (Z_m) = \sum_{m \ge 1} \ln |Z_m| + i \sum_{... | joriki | 6,622 | <p>The problem is ill-posed. You answered one possible interpretation of it, but apparently another interpretation is intended. To say that a child is chosen “randomly” tells us nothing without specifying the distribution according to which it is chosen. Typically, when no distribution is specified, the int... |
1,059,427 | <p>What is a good method to number of ways to distribute $n=30$ distinct books to $m=6$ students so that each student receives at most $r=7$ books?</p>
<p>My observation is: If student $S_i$ receives $n_i$ books, the number of ways
is: $\binom{n}{n_1,n_2,\cdots,n_m}$.</p>
<p>So answer is coefficient of $x^n$ in $n!(... | leonbloy | 312 | <p>Not very easy, but calling $S(b,s,m)$ the number of distributions ($b$ books, $s$ students, $m$ maximum allowed for each) one could write:</p>
<p>$$S(b,s,m)= \sum_{k=0}^{\min(\lfloor b/m \rfloor,s)} {s \choose k} \frac{b!}{(m!)^k(b-m \,k)!} S(b-mk,s-k,m-1) $$</p>
<p>and compute recursively the values, with $S(b,s,... |
102,814 | <p>Is it possible to construct a nontrivial homomorphism from $C_6$ to $A_3$? I have tried to construct one but failed. Is there a good way to see when there will be a homomorphism?</p>
| Community | -1 | <p>I have the following recipe in mind:</p>
<p><strong>Proposition:</strong></p>
<p>The number of homomorphisms from $\mathbb Z_m$ to $\mathbb Z_n$ is $(m,n)$.</p>
<p><em>Proof.</em> </p>
<p>Observe that a homomorphism from a cyclic group is fixed by fixing the image of a generator, $1$. </p>
<p>$$\varphi(1)=a \im... |
1,960,911 | <p>I am trying to evaluate this limit for an assignment.
$$\lim_{x \to \infty} \sqrt{x^2-6x +1}-x$$</p>
<p>I have tried to rationalize the function:
$$=\lim_{x \to \infty} \frac{(\sqrt{x^2-6x +1}-x)(\sqrt{x^2-6x +1}+x)}{\sqrt{x^2-6x +1}+x}$$</p>
<p>$$=\lim_{x \to \infty} \frac{-6x+1}{\sqrt{x^2-6x +1}+x}$$</p>
<p>Th... | adjan | 219,722 | <p>Your error is here:
$$\frac{\sqrt{x^2-6x +1}-x}{x}=\sqrt{1-\frac{6}{x}+\frac{1}{x^2}}+1$$</p>
|
3,322,049 | <p>I could have solved this by substitution, but the ‘n’ is confusing me. How should I proceed?</p>
| J.G. | 56,861 | <p>If the question really means <span class="math-container">$\int\frac{dx}{x\ln n}$</span> with <span class="math-container">$n$</span> a constant with respect to <span class="math-container">$x$</span>, you get <span class="math-container">$\frac{\ln |x|}{\ln n}+C=\log_n |x|+C$</span>. If it's a typo for <span class=... |
3,322,049 | <p>I could have solved this by substitution, but the ‘n’ is confusing me. How should I proceed?</p>
| st.math | 645,735 | <p>If you meant '<span class="math-container">$x$</span>' instead of '<span class="math-container">$n$</span>', then substituting <span class="math-container">$u=\ln x$</span> gives <span class="math-container">$\mathrm du=\frac1x\,\mathrm dx$</span>, and thus
<span class="math-container">$$\int\frac{1}{x\ln x}\mathrm ... |
253,966 | <p>Just took my final exam and I wanted to see if I answered this correctly:</p>
<p>If $A$ is a Abelian group generated by $\left\{x,y,z\right\}$ and $\left\{x,y,z\right\}$
have the following relations:</p>
<p>$7x +5y +2z=0; \;\;\;\; 3x +3y =0; \;\;\;\; 13x +11y +2z=0$</p>
<p>does it follow that $A \cong Z_{3} \tim... | DonAntonio | 31,254 | <p>Have you studied The Smith Normal Form of an (integer) square matrix? Well, if you form the matrix of coefficients of your relations you get:</p>
<p>$$A:=\begin{pmatrix}7&5&2\\3&3&0\\13&11&2\end{pmatrix}\Longrightarrow \det A=0$$</p>
<p>Thus, if $\,G:=\{x,y,z\}\,$ is the free abelian group ... |
119,876 | <pre><code>Module[{x},
f@x_ = x;
p@x_ := x;
{x, x_, x_ -> x, x_ :> x}
]
?f
?p
</code></pre>
<p>gives</p>
<pre><code>{x$17312, x$17312_, x_ -> x, x_ :> x}
f[x_]=x
p[x_]:=x
</code></pre>
<p>but I'd like to get</p>
<pre><code>{x$17312, x$17312_, x$17312_ -> x$17312, x$17312_ :> x$17312}
f[x$17312... | Alexey Popkov | 280 | <p>It looks like what you need is the <a href="http://library.wolfram.com/infocenter/MathSource/425/" rel="nofollow"><code>LocalPatterns`</code></a> package by Ted Ersek. It introduces new special symbols <code>DotEqual</code> and <code>LongRightArrow</code> which works like <code>Set</code> and <code>Rule</code> but p... |
402,802 | <p>I have read that $$y=\lvert\sin x\rvert+ \lvert\cos x\rvert $$ is periodic with fundamental period $\frac{\pi}{2}$.</p>
<p>But <a href="http://www.wolframalpha.com/input/?i=y%3D%7Csinx%7C%2B%7Ccosx%7C" rel="nofollow">Wolfram</a> says it is periodic with period $\pi$.</p>
<p>Please tell what is correct.</p>
| Matt L. | 70,664 | <p><strong>Hint:</strong> Note that $\sin(x+\pi/2) = \cos(x)$ and $\cos(x+\pi/2)=-\sin(x)$.</p>
|
402,802 | <p>I have read that $$y=\lvert\sin x\rvert+ \lvert\cos x\rvert $$ is periodic with fundamental period $\frac{\pi}{2}$.</p>
<p>But <a href="http://www.wolframalpha.com/input/?i=y%3D%7Csinx%7C%2B%7Ccosx%7C" rel="nofollow">Wolfram</a> says it is periodic with period $\pi$.</p>
<p>Please tell what is correct.</p>
| mez | 59,360 | <p>You should try this and figure it out. It is not that hard, and it is a bad habbit to give up on something that you can do. Wolfram alpha is not always correct, it's written by humans.</p>
<p><img src="https://i.stack.imgur.com/NHOcv.jpg" alt="enter image description here"></p>
<p>It's true that this function is $... |
481,834 | <p>Let $A=(A_{ij})$ be a square matrix of order $n$. Verify that the determinant of the matrix</p>
<p>$\left( \begin{array}{ccc}
a_{11}+x & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22}+x & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{n1} & a_{n2} & \cdots & ... | Eurakarte | 92,668 | <p>Simply note that a determinant is computed as the sum and product of finitely many polynomial terms, which results in a polynomial. You can show that with an induction argument, although I don't really think that's necessary.</p>
|
2,906,917 | <p>I have this problem but I don't know how to continue.<br>
Here it is:
Compute $\int \sin(x) \left( \frac{1}{\cos(x) + \sin(x)} + \frac{1}{\cos(x) - \sin(x)} \right)\,dx.$<br>
So I can anti differentiate the sin x to be cos x but I am unsure on where to go off that for the fraction. I don't want to multiply the fract... | YiFan | 496,634 | <p>Multiplying the fractions is actually the way to go and everything cancels out!
$$\int\sin x\left(\frac{2\cos x}{\cos^2x-\sin^2x}\right)dx = \int\frac{\sin 2x}{\cos 2x}dx = \int\tan 2x\; dx$$
Now, you just need to find the antiderivative of $\tan$. </p>
<p>By the way: partial fraction decomposition is only done for... |
2,648,516 | <p>I am studying Fourier analysis from the text "Stein and Shakarchi" and there is this thing on Dirichlet Kernel. It's fine to define it as a trigonometric poylnomial of degree $n$ , but what is the mathematical intuition behind calling it a Kernel ? I have also thought of Kernel as being a set of zeroes of sum functi... | Disintegrating By Parts | 112,478 | <p>The "kernel" of something is the essential part of it, the germ of it, a whole seed. That's a definition that makes sense in terms of the linear operator $T$, because the essential part of it is the function $K$. To me, it makes no sense how the null space came to be called "kernel." I would guess that the use of ke... |
2,179,317 | <p>We know that, if $\mathcal D$ is a domain containing the origin $(0,0,0)$, then</p>
<p>$$\int_{\mathcal D} \delta(\vec r) d \vec r= \int_{\mathcal D} \delta(x) \delta(y) \delta(z) dx dy dz=1$$</p>
<p>However, <a href="http://mathworld.wolfram.com/DeltaFunction.html" rel="nofollow noreferrer">we also know that</a> ... | Stefano | 387,021 | <p>I would be inclined to say that your intuition is wrong, mostly because $\delta$ is not a function. </p>
<p>I'll explain. Take the function $\delta_n$ defined by $\delta_n(x) = n$ if $x \in (-1/n,0)$ and $\delta_n(x) =0$ otherwise. Clearly $\delta_n(x) = 0$ for $x \ge 0$, and so
$$
\int_0^\infty \delta_n(x) = 0
$$
... |
1,507,710 | <p>I'm trying to get my head around group theory as I've never studied it before.
As far as the general linear group, I think I've ascertained that it's a group of matrices and so the 4 axioms hold?
The question I'm trying to figure out is why $(GL_n(\mathbb{Z}),\cdot)$ does not form a group.
I think I read somewhere... | Geoff Robinson | 13,147 | <p>As your title suggests, ${\rm GL}(n, \mathbb{Z})$ is indeed a group. It consists of those integer matrices with non-zero determinant whose inverses are also integer matrices ( and such matrices all have determinant $\pm 1$, as others have pointed out). </p>
<p>What is not a group is the set of $n \times n$ integer ... |
1,507,710 | <p>I'm trying to get my head around group theory as I've never studied it before.
As far as the general linear group, I think I've ascertained that it's a group of matrices and so the 4 axioms hold?
The question I'm trying to figure out is why $(GL_n(\mathbb{Z}),\cdot)$ does not form a group.
I think I read somewhere... | testman | 286,006 | <p>Integers with multiplication do not form a group. For example the 1x1 matrix (2) has an inverse (1/2) which is not integer.</p>
|
66,199 | <p>Say I have the following lists of rules:</p>
<pre><code>case1 = {a -> 1, b -> 3, c -> 4, e -> 5}
case2 = {c -> 3, a -> 1, w -> 2}
case3 = {x -> 5, y -> 2, z -> 0, c -> 2}
</code></pre>
<p>How do I write a function <code>myfun[]</code>, to select the value of "c" in each case?</p>
... | mgamer | 19,726 | <p>In such a case, with Mathematica 10, one can use easily associations:</p>
<pre><code>case1 = {a -> 1, b -> 3, c -> 4, e -> 5}
case2 = {c -> 3, a -> 1, w -> 2}
case3 = {x -> 5, y -> 2, z -> 0, c -> 2}
</code></pre>
<p>then: </p>
<pre><code>ac1 = Association@case1;
ac2 = Association... |
3,410,150 | <p>If we try solving it by finding <span class="math-container">$f''(x)$</span> then it is very long and difficult to do, so my teacher suggested a way of doing it, he said find nature of all the roots of <span class="math-container">$f(x) =f'(x)$</span>, and on finding nature of the roots we got them to be real(but no... | Community | -1 | <p><strong>Answer to subsidiary question</strong></p>
<p>When you say <span class="math-container">$f'(x)=f(x)$</span> you simply mean that they are equal for some specific roots. That does not mean their derivatives are equal.</p>
|
528,591 | <p>I need to prove there are zero divisors in $\mathbb{Z}_n$ if and only if $n$ is not prime.
What should I consider first? </p>
| esoteric-elliptic | 425,395 | <p>Here's a slightly different way to go about this: to prove that <span class="math-container">$\mathbb Z_n$</span> has zero divisors if and only if <span class="math-container">$n$</span> is not prime, is the same as showing that <span class="math-container">$\mathbb Z_n$</span> has no zero divisors if and only if <s... |
458 | <p>If you go to the bottom of any page in the SE network (e.g. this one!), you'll see a list of SE sites. In particular there's a link to MathOverflow, that is potentially seen by a large number of people (many of whom are outside of our target audience).</p>
<p>When you put your cursor over that link, there's a hover... | Kaveh | 7,507 | <p>Professional Researchers in Mathematics</p>
|
1,383,956 | <p>Having two points <span class="math-container">$A(xa, ya)$</span> and <span class="math-container">$B(xb, yb)$</span> and knowing a value <span class="math-container">$k$</span> representing the length of a perpendicular segment in the middle of <span class="math-container">$[AB]$</span>, how can I find the other po... | 3SAT | 203,577 | <p>After using @Hetebrij answer</p>
<p>$\color{blue}{(4)}$</p>
<p>$P(X> 0.5)=\displaystyle\int_{0.5}^{\ln(2)}=0.351\;,P(X\geq 0.3)=0.651$</p>
<p>$P(X\geq 0.5\big|X\geq 0.3)=\boxed{\frac{0.351}{0.651}\approx 0.539}$</p>
<hr>
<p>$P(X>0.7)=\boxed0$ because that the max height is $\ln(2)\approx 0.69$</p>
|
441,374 | <p>Let $K_{\alpha}(z)$ be the <a href="https://en.wikipedia.org/wiki/Bessel_function#Modified_Bessel_functions:_I.CE.B1_.2C_K.CE.B1" rel="nofollow noreferrer">modified Bessel function of the second kind of order $\alpha$</a>.</p>
<p>I need to compute the following integral:</p>
<p>$$\int_0^\infty\;\;K_0\left(\sqrt{a(... | Kirill | 11,268 | <p>Using some of the ideas mentioned in Ron Gordon's answer we can evaluate this integral. Change the variable to $x=\sqrt{a} k$, so that the integral becomes
$$ \int_0^\infty K_0(\sqrt{ak^2+ab})\,dk = \frac1{\sqrt{a}} \int_0^\infty K_0(\sqrt{x^2+ab})\,dx, $$
and introduce the function
$$ I(b) = \int_0^\infty K_0(\sqrt... |
1,837,807 | <p>Let $\mathbb{N}$ denote the set of natural numbers, then a subbasis on $\mathbb{N}$ is </p>
<p>$$S = \{(-\infty, b), b \in \mathbb{N}\} \cup \{(a,\infty), a \in \mathbb{N}\}$$</p>
<p>Let $\leq$ be the relation on $\mathbb{N}$ identified with "less or equal to"</p>
<p>Then I saw a claim that says: (the order topol... | Community | -1 | <p>You are confusing</p>
<p>$$\{(-\infty, b) \mid b \in \mathbb{N}\} \cup \{(a,\infty) \mid a \in \mathbb{N}\}$$</p>
<p>with</p>
<p>$$ \{ (-\infty, b) \cup (a, \infty) \mid b \in \mathbb{N}, a \in \mathbb{N} \}$$</p>
|
3,644,710 | <p>I wish to classify the Galois group of <span class="math-container">$\mathbb{Q}(e^{i\pi/4})/\mathbb{Q}$</span>. Let me denote the eighth root of unity as <span class="math-container">$\epsilon$</span>. I see that <span class="math-container">$1, \epsilon, \epsilon^2, \epsilon^3$</span> are linearly independent over ... | Itachi | 655,021 | <p>Hint :
Firstly, go for m>n and use <span class="math-container">$|x_m - x_n| \le |x_m - x_{m-1}|+|x_{m-1} - x_{m-2}|+....+|x_{n+1} - x_n|$</span>.
Now plug in all the values.</p>
|
413,719 | <p>Would I be correct in saying that they correspond to all points in $\mathbb{R}^3$? Or a line in $\mathbb{R}^3$?</p>
| Sujaan Kunalan | 77,862 | <p>They are just vectors in $\mathbb{R}^3$. They do not correspond to all points nor do they make a line.</p>
|
1,010 | <p>For periodic/symmetric tilings, it seems somewhat "obvious" to me that it just comes down to working out the right group of symmetries for each of the relevant shapes/tiles, but its not clear to me if that carries over in any nice algebraic way for more complicated objects such as a <a href="http://en.wikipedia.org/... | Michael Lugo | 143 | <p>This reminds me of a talk that I saw by <a href="http://www.ma.utexas.edu/~sadun" rel="nofollow">Lorenzo Sadun</a> a couple years ago, although I'm not sure exactly why. (When I think about tilings I'm thinking more combinatorially than it sounds like you are.) You might look at <a href="http://www.ma.utexas.edu/u... |
1,010 | <p>For periodic/symmetric tilings, it seems somewhat "obvious" to me that it just comes down to working out the right group of symmetries for each of the relevant shapes/tiles, but its not clear to me if that carries over in any nice algebraic way for more complicated objects such as a <a href="http://en.wikipedia.org/... | Emily Peters | 699 | <p>In the answers to "what is a groupoid" I was pointed to
<a href="http://www.ams.org/notices/199607/weinstein.pdf">Alan
Weinstein's very nice notices article</a>.</p>
<p>The first example he gives (before the definition even of a groupoid)
is about tilings, and how the groupoid contains more information than
the ... |
3,689,627 | <p>First of all let me tell you that the answer to this question is likely to confirm a not-so-minor error in a very popular (and excellent) textbook on optimization, as you'll see below. </p>
<h3>Background</h3>
<p>Suppose that we have a real-valued function <span class="math-container">$f(X)$</span> whose domain is... | greg | 357,854 | <p>Consider a pair matrices with elements given by
<span class="math-container">$$\eqalign{
M_{ij} &= \begin{cases} 1 &\text{if }(i=j) \\
\frac{1}{2} & \text{otherwise}\end{cases} \\
W_{ij} &= \begin{cases} 1 &\text{if }(i=j) \\
2 & \text{otherwise}\end{cases} \\
}$$</span>
which are Hadamard... |
3,689,627 | <p>First of all let me tell you that the answer to this question is likely to confirm a not-so-minor error in a very popular (and excellent) textbook on optimization, as you'll see below. </p>
<h3>Background</h3>
<p>Suppose that we have a real-valued function <span class="math-container">$f(X)$</span> whose domain is... | Miguel | 259,671 | <p>I think that the key problem is that such differential on "sets of matrices with dependent components" is not defined.</p>
<p>If <span class="math-container">$f:\mathbb{R}^m \rightarrow \mathbb{R}$</span> is differentiable, then the first order approximation in the direction of <span class="math-container">$v$</spa... |
799,183 | <p>I'm trying to work out what the transformation $T:z \rightarrow -\frac{1}{z}$ does (eg reflection in a line, rotation around a point etc). Any help on how to do this would be greatly appreciated! I've tried seeing what it does to $1$ and $i$ but is hasn't helped me. Thanks!</p>
| Alice Ryhl | 132,791 | <p>It's a <a href="http://en.wikipedia.org/wiki/Inversive_geometry#Circle_inversion" rel="nofollow">circle inversion</a>, followed by a reflection over the y axis.</p>
|
1,477,325 | <p>It is known that an integrable function is a.e. finite. Is an a.e. finite function integrable? What if the measure is finite?</p>
| Paul | 202,111 | <p>No. A characteristic function of a non measurable set is everywhere finite, but not integrable.</p>
|
3,368,402 | <p>I am utilizing set identities to prove (A-C)-(B-C).</p>
<p><span class="math-container">$\begin{array}{|l}(A−B)− C = \{ x | x \in ((x\in (A \cap \bar{B})) \cap \bar{C}\} \quad \text{Def. of Set Minus}
\\
\quad \quad \quad \quad \quad =\{ x | ((x\in A) \wedge (x\in\bar{B})) \wedge (x\in\bar{C})\} \quad \text{Def. o... | user0102 | 322,814 | <p><span class="math-container">\begin{align*}
(A-C)-(B-C) & = (A\cap\overline{C})-(B\cap\overline{C}) = (A\cap\overline{C})\cap(\overline{B\cap\overline{C})}\\\\
& = (A\cap\overline{C})\cap(\overline{B}\cup C) = (A\cap\overline{C}\cap\overline{B})\cup(\overline{C}\cap\overline{B}\cap C)\\\\
& = A\cap\overl... |
3,492,376 | <p>Can anyone explain to me why the below expression:</p>
<p><span class="math-container">$$\int\frac{2\cos x}{{(4-4\sin^2x})^{3/2}}\:dx$$</span></p>
<p>is equal to this:</p>
<p><span class="math-container">$$\frac{2}{8}\int\frac{\cos x}{{(1-\sin^2x})^{3/2}}\:dx$$</span></p>
<p>a) Why the constant <span class="math... | Michael Hardy | 11,667 | <p><span class="math-container">$$
4^{3/2} = \sqrt 4^3 = 2^3 = 8.
$$</span></p>
|
114,371 | <p>A sector $P_1OP_2$ of an ellipse is given by angles $\theta_1$ and $\theta_2$. </p>
<p><img src="https://i.stack.imgur.com/mdmq2.png" alt="A sector of an ellipse"></p>
<p>Could you please explain me how to find the area of a sector of an ellipse?</p>
| Riccardo.Alestra | 24,089 | <p>Using polar elliptical coordinates:$$x=a\rho cos(\theta)$$
$$y=b \rho sin(\theta)$$
the part of the plane enclosed from the ellipse is
$$\{(\rho,\theta):0\le \rho \le 1,\theta\in(0,2\pi)\}$$
the Jacobian of the inverse transform is:
$$J=\begin{bmatrix}a cos(\theta) & -a \rho sin(\theta) \\ b sin(\theta) & b ... |
114,371 | <p>A sector $P_1OP_2$ of an ellipse is given by angles $\theta_1$ and $\theta_2$. </p>
<p><img src="https://i.stack.imgur.com/mdmq2.png" alt="A sector of an ellipse"></p>
<p>Could you please explain me how to find the area of a sector of an ellipse?</p>
| Community | -1 | <p>As I show in <a href="https://math.stackexchange.com/questions/493104">Evaluating $\int_a^b \frac12 r^2\ \mathrm d\theta$ to find the area of an ellipse</a> the area
of an ellipse given its central angle is: </p>
<p><span class="math-container">$A(\theta) = \frac{1}{2} a b \tan ^{-1}\left(\frac{a \tan (\theta )}{b... |
65,886 | <p>It is clear that Sylow theorems are an essential tool for the classification of finite groups.
I recently read an article by Marcel Wild, <em>The Groups of Order Sixteen Made Easy</em>, where he gives a complete classification of the groups of order $16$ that is based on
elementary facts, in particular, he does not ... | Panurge | 72,877 | <p>Thanks to user641 for this concise proof. If I'm not wrong, it is a variant of Cole's proof (1893), which can be found here :</p>
<p><a href="http://www.jstor.org/stable/2369516?seq=1#page_scan_tab_contents" rel="nofollow noreferrer">http://www.jstor.org/stable/2369516?seq=1#page_scan_tab_contents</a></p>
<p>Perha... |
815,065 | <p>$$\int^{\pi /2}_{0} \frac{\ln(\sin x)}{\sqrt x}dx$$</p>
<p>Use the segment integral formula? The $\sqrt x$ is zero at $x=0$ and $\ln\sin x$ is $-\infty$ </p>
| Lucian | 93,448 | <p>We know that $\displaystyle\underbrace{\int_0^\tfrac\pi2\frac{\ln x}{\sqrt x}dx}_A=\sqrt{2\pi}\bigg(\ln\frac\pi2-2\bigg)$. This can easily be proven using integration by </p>
<p>parts. Now, let's show that $\displaystyle\int_0^\tfrac\pi2\frac{\ln(\sin x)}{\sqrt x}dx$ is bounded : $\displaystyle\int_0^\tfrac\pi2\fra... |
372,548 | <p><span class="math-container">$f:\mathbb R\to\mathbb R$</span> is a convex continuous function. We have a finite or a countable set of triples: <span class="math-container">$\{(x_n,f(x_n),D_n)\}_{n\in N}$</span>, where <span class="math-container">$D_n$</span> is the slope of a tangent line <span class="math-containe... | Willie Wong | 3,948 | <p>The local-in-time regularity of the Navier-Stokes solution is pretty well-studied.</p>
<ol>
<li>The classic paper of Foias and Temam <a href="https://www.sciencedirect.com/science/article/pii/0022123689900153" rel="nofollow noreferrer">https://www.sciencedirect.com/science/article/pii/0022123689900153</a> proves tha... |
690,465 | <p>So we are learning trigonometry in school and I would like to ask for a little help with these. I would really appreciate if somebody can explain me how I can solve such equations :)</p>
<ul>
<li><p>$\sin 3x \cdot \cos 3x = \sin 2x$</p></li>
<li><p>$2( 1 + \sin^6 x + \cos^6 x ) - 3(\sin^4 x + \cos^4 x) - \cos x = 0... | Henry | 6,460 | <p>Are your formulae things like $\sin^2 x + \cos^2 x = 1$?</p>
<p>If so then you need to spot where you can apply them.</p>
<p>For example $$\sin^2 x - \sin^4 x + \cos^4 x = 1$$
$$\sin^2 x (1- \sin^2 x) + \cos^4 x = 1$$
$$\sin^2 x \cos^2 x + \cos^4 x = 1 $$
$$(\sin^2 x + \cos^2 x) \cos^2 x = 1 $$
$$ \cos^2 x = ... |
690,465 | <p>So we are learning trigonometry in school and I would like to ask for a little help with these. I would really appreciate if somebody can explain me how I can solve such equations :)</p>
<ul>
<li><p>$\sin 3x \cdot \cos 3x = \sin 2x$</p></li>
<li><p>$2( 1 + \sin^6 x + \cos^6 x ) - 3(\sin^4 x + \cos^4 x) - \cos x = 0... | Raven | 130,125 | <p>For the third one
$$3\sin^2x-4\sin x\cos x+5\cos^2x=2\\
\Rightarrow\sin^2x-4\sin x\cos x+4\cos^2x=2-2\sin^2x-\cos^2x\\
\Rightarrow(\sin x-2\cos x)^2=2(1-\sin^2x)-\cos^2x\\
\Rightarrow(\sin x-2\cos x)^2=2\cos^2x-\cos^2x\\
\Rightarrow(\sin x-2\cos x)^2=cos^2x\\
\Rightarrow(\sin x-2\cos x)^2-\cos^2x=0\\
\Rightarrow(\si... |
4,485,550 | <p>I'm interested in playing with nonwell-founded variants of set theory and weaker/different axioms of induction/extensionality.</p>
<p>I have a hunch coalgebraic methods could better handle weirdness like modelling homotopy type theory.</p>
<p>I also have just been interested in the idea of coinduction as primitive a... | Greg Nisbet | 128,599 | <p>Here is one entrypoint to a particular family of non-well-founded set theories, New Foundations and its descendants.</p>
<p>All variants of New Foundations have a universal set <span class="math-container">$U$</span> and have <span class="math-container">$U \in U$</span> as a theorem, so none of them are well-founde... |
2,202,382 | <p>When $A^TA = I$, I am told it is orthogonal. What does that mean?</p>
<p>$A = \begin{bmatrix}cos\theta & & -sin\theta \\ \\ sin\theta & & cos\theta\end{bmatrix}, A^T = \begin{bmatrix}cos\theta & & sin\theta \\ \\ -sin\theta & & cos\theta\end{bmatrix}$</p>
| PM. | 416,252 | <p>Perhaps the term is <em>suggestive</em> of matrix $\mathbf{A}$ being orthogonal to another matrix $\mathbf{B}$ in some sense like $$\mathbf{A}.\mathbf{B}=0$$ as it would be if $A$ and $B$ were orthogonal vectors. However this is <em>not</em> what it means when considering matrices.</p>
<p>For matrices ìt refers to ... |
1,939,937 | <p>$(172195)(572167)=985242x6565$</p>
<p>Obviously the answer is 9 if you have a calculator, but how can you find x without redoing the multiplication?</p>
<p>The book says to use congruences, but I don't see how that is very helpful. </p>
| fleablood | 280,126 | <p>Old trick.</p>
<p>$X = \sum_{i=0}^na_i 10^i = \sum{i=0}^n a_i (9 + 1)^i \equiv \sum_{i=0}^n a_i \mod 9$.</p>
<p>This is why adding up the digits of a number will give you the remainder of the number when dividing by 9[*].</p>
<p>So $172195 \equiv 1+7+2+1+9+5 \equiv 7 \mod 9$</p>
<p>And $572167 \equiv 5+7+2+1+6+7... |
4,804 | <p>Using the Unanswered tab, I was surprised to find a large number of questions that were already answered <em>in the answer box</em>, correctly, and thoroughly. But if the answer(s) is never upvoted, the question remains in the tab where it does not belong. To make things worse, the Community user occasionally bumps ... | Amitesh Datta | 10,467 | <p>The reason that a user might not upvote good answers to his/her question is usually that he/she does not know how to, forgets to do so, or does not notice the answers. Of course, this most often happens when the user is new/unregistered.</p>
<p>However, you may also ask why other users do not upvote good answers. T... |
4,804 | <p>Using the Unanswered tab, I was surprised to find a large number of questions that were already answered <em>in the answer box</em>, correctly, and thoroughly. But if the answer(s) is never upvoted, the question remains in the tab where it does not belong. To make things worse, the Community user occasionally bumps ... | 75064 | 75,064 | <p>Users interested in reducing the backlog of Unanswered questions by upvoting helpful answers may find the following queries useful:</p>
<ul>
<li><p><a href="http://data.stackexchange.com/mathematics/query/114659/thank-you-comment-by-op-without-upvote-or-accept" rel="nofollow">Thank-you comment by OP on the sole ans... |
1,753,719 | <p>Definition of rapidly decreasing function</p>
<p>$$\sup_{x\in\mathbb{R}} |x|^k |f^{(l)}(x)| < \infty$$ for every $k,l\ge 0$.</p>
<p>Given the Gaussian function $f(x) = e^{-x^2}$, I know that its derivatives will always be in form of $P(x)e^{-x^2}$ where $P(x)$ is a polynomial of degree, say, $n$. Then $|x|^k |f... | Robert Israel | 8,508 | <p>An exponential always beats a polynomial in the end...</p>
<p>$|x| \le e^{|x|}$ so $|x|^n < e^{n |x|} < e^{x^2}$ if $|x| > n$, therefore $\left|x^n e^{-x^2}\right| < 1$ there. </p>
<p>Since the continuous function $x^n e^{-x^2}$ is also
bounded on the finite interval $[-n,n]$, we conclude that $x^n e... |
97,946 | <p>I want to prove the following:</p>
<p>Let $G$ be a finite abelian $p$-group that is not cyclic.
Let $L \ne {1}$ be a subgroup of $G$ and $U$ be a maximal subgroup of L then there exists a maximal subgroup $M$ of $G$ such that $U \leq M$ and $L \nleq M$.</p>
<p>Proof.
If $L=G$ then we are done.Suppose $L \ne G$ . ... | yakov | 92,209 | <p>A s noted Prof. Robinson, this is false. However, this is true iff $L\not\le M\Phi(G)$. Indeed, let $\bar G=G/M\Phi(G)$; then $\bar L$ is a direct factor of $\bar G$ of order $p$. If $\bar G=\bar L\times\bar U$. Then $U$ is a maximal subgroup of $G$ and $U\cap L=M$.</p>
|
1,443,680 | <p>In Quantum Mechanics one often deals with wavefunctions of particles. In that case, it is natural to consider as the space of states the space $L^2(\mathbb{R}^3)$. On the other hand, on the book I'm reading, there's a construction which it's quite elegant and general, however it is not rigorous. For those interested... | Physics Footnotes | 348,696 | <p>There are two mathematically rigorous (and quite general) Hilbert Space formalisms you might be looking for. Both can be seen as attempts to salvage the engine of Dirac's original bra-ket algorithm, while avoiding its mathematical embarrassments.</p>
<p><strong>The first</strong> - created by von Neumann - replaces... |
19,962 | <p><a href="http://en.wikipedia.org/wiki/Covariance_matrix" rel="nofollow">http://en.wikipedia.org/wiki/Covariance_matrix</a></p>
<pre><code>Cov(Xi,Xj) = E((Xi-Mi)(Xj-Mj))
</code></pre>
<p>Is the above equivalent to:</p>
<pre><code>(Xi-Mi)(Xj-Mj)
</code></pre>
<p>I don't understand why the expectancy of (Xi-Mi)(Xj-... | leonbloy | 312 | <p>It's like asking if $X$ is equivalent to $E(X)$. </p>
<p>$X$ is (asssumed to be) a random variable, it can take several values according to some probability law. In contrast $E(X)$ is the expected value of $X$ - so it's not a random variable which takes different values in different tries, it's a constant number. ... |
2,772,541 | <p>I have a given function:$(x_i, y_i) = p(x_a, y_a) + (1-p)(x_b, y_b)$</p>
<p>From this function I need to get $p$, which should be a value between 0 and 1. </p>
<p>In an attempt to solve this I found: </p>
<p>$(x_i, y_i) = p(x_a, y_a) + (1-p)(x_b, y_b)$</p>
<p>$(x_i, y_i) - (x_b, y_b) = p(x_a, y_a) - p(x_b, y_b)$... | Hagen von Eitzen | 39,174 | <p>Let $u,v\in(a,b)$ be zeroes of $f$ and assume that $g$ has no zeroes in $(u,v)$.
From $f(u)=f(v)=0$ and the given inequality, we obtain that $g(u)$, $g(v)$ are non-zero, hence also positive.
This allows us to consider $h(x)=\frac{f(x)}{g(x)}$ on $[u,v]$.
We have $h(u)=h(v)=0$. By Rolle $h'(x)=0$ for some $x\in(u,v)... |
2,772,541 | <p>I have a given function:$(x_i, y_i) = p(x_a, y_a) + (1-p)(x_b, y_b)$</p>
<p>From this function I need to get $p$, which should be a value between 0 and 1. </p>
<p>In an attempt to solve this I found: </p>
<p>$(x_i, y_i) = p(x_a, y_a) + (1-p)(x_b, y_b)$</p>
<p>$(x_i, y_i) - (x_b, y_b) = p(x_a, y_a) - p(x_b, y_b)$... | Chappers | 221,811 | <p>Let $W = f' g - g' f$. Then $W \neq 0$ by the condition, and if $x_1,x_2$ are successive zeros of $f$, then
$$ W(x_1) = f'(x_1) g(x_1), \qquad W(x_2) = f'(x_2) g(x_2), $$
and these both have the same sign. But $f'(x_1)f'(x_2)<0$ because otherwise the mean value theorem and the intermediate value theorem would im... |
2,156,331 | <p>Consider the discrete topology $\tau$ on $X:= \{ a,b,c, d,e \}$. Find subbasis for $\tau$ which does not contain any singleton sets.</p>
<p>The definition of subbasis is as follows: </p>
<blockquote>
<p><strong>Definition:</strong> A <em>subbasis</em> $S$ for a topology on $X$ is a collection of subsets of $X$ w... | MPW | 113,214 | <p><strong>Hint:</strong> You can write $\{a\}$ as $\{a,b\}\cap\{a,c\}$. Do the same with each of the elements of $X$.</p>
|
1,256,460 | <p>I want to solve the following problem: </p>
<p>$$u_{xx}(x,y)+u_{yy}(x,y)=0, 0<x<\pi, y>0 \\ u(0,y)=u(\pi, y)=0, y>0 \\ u(x,0)=\sin x +\sin^3 x, 0<x<\pi$$ </p>
<p>$u$ bounded </p>
<p>I have done the following: </p>
<p>$$u(x,y)=X(x)Y(y)$$ </p>
<p>We get the following two problems: </p>
<p>$$X''... | Valentin | 223,814 | <p>Basically, by just using definition of average</p>
<p>$T_{avg}=\frac{1}{30}\int_0^{30} { T(t) dt }$ </p>
|
908,196 | <blockquote>
<p>Solve $x^2-1=2$</p>
</blockquote>
<p>I have no idea how to do this can somebody please help me? I have tried working it out and I could never get the answer.</p>
| Ahaan S. Rungta | 85,039 | <p>$ x^2 - 1 = 2 \implies x^2 = 3 \implies x = \pm \sqrt{3} $</p>
<p>We have two solutions because both solutions result in the same square. We simply added $1$ to both sides and took the square root. Both steps are correct, because we are making the same transformation to both sides of the equation. So if they are eq... |
908,196 | <blockquote>
<p>Solve $x^2-1=2$</p>
</blockquote>
<p>I have no idea how to do this can somebody please help me? I have tried working it out and I could never get the answer.</p>
| John Joy | 140,156 | <p>$$\begin{array}{llll}
x^2-1&=&2 &(\text{given})\\
(x^2-1)+1&=&2+1 &(\text{additive axiom of equality})\\
x^2+(-1+1)&=&3 &(\text{associative field axiom})\\
x^2+0&=&3 &(\text{additive inverse field axiom})\\
x^2&=&3 &(\text{additive identity})
\end{array}$$<... |
745,436 | <p>I'm reading this pdf <a href="http://rutherglen.science.mq.edu.au/wchen/lndpnfolder/dpn01.pdf" rel="nofollow">http://rutherglen.science.mq.edu.au/wchen/lndpnfolder/dpn01.pdf</a> I understand some of the expression used in this but I don't understand the part $(m,n) = 1$</p>
<p>Is this a cartesian coordinate or some... | Bill Dubuque | 242 | <p>The notation $\,(a,b) := \gcd(a,b)\,$ is widely used in number theory. Similarly, but less frequently, authors use $\,\ \ [a,b]\, := {\rm lcm}(a,b).\,$ Hence $\,(a,b) = 1\,$ means $\,a,b\,$ are <em>coprime:</em> $\,c\mid a,b\,\Rightarrow\,c\mid 1.$</p>
<p>Here, as often, one can uniquely infer the meaning from it... |
886,003 | <p>I have two questions:</p>
<p><strong>A)</strong> Suppose that we have $$Z=c\sum_i (X_i-a)(Y_i-b)$$ where $X_i$s and $Y_i $s are independent exponential random variables with means equal to $\mu_{X}$ and $\mu_{Y}$ (for $1\le i\le n$). That is $X_i$s are i.i.d random variables and so are $Y_i $s. Besides, $a,b$ and ... | wolfies | 74,360 | <p>In answer to your first question ... </p>
<p>Given <span class="math-container">$X \sim Exponential(\lambda_1)$</span> with <span class="math-container">$E[X] =\lambda_1 $</span>, and <span class="math-container">$Y \sim Exponential(\lambda_2)$</span> with <span class="math-container">$E[Y] =\lambda_2 $</span>, whe... |
499,652 | <p>I saw this a lot in physics textbook but today I am curious about it and want to know if anyone can show me a formal mathematical proof of this statement? Thanks!</p>
| Daniel Robert-Nicoud | 60,713 | <p>As already expressed in another answer, this can be formulated in a formal way by taking the Taylor expansion of $\tan\alpha$ around $0$ to get:
$$\tan\alpha = \alpha + \frac{1}{3}\alpha^3 + \ldots = \alpha + O(\alpha^3)$$
where $O(\alpha^3)$ denotes some function that goes to zero approximately at the same rate as ... |
2,761,151 | <p>In the formula below, where does the $\frac{4}{3}$ come from and what happened to the $3$? How did they get the far right answer? Taken from Stewart Early Transcendentals Calculus textbook.</p>
<p>$$\sum^\infty_{n=1} 2^{2n}3^{1-n}=\sum^\infty_{n=1}(2^2)^{n}3^{-(n-1)}=\sum^\infty_{n=1}\frac{4^n}{3^{n-1}}=\sum_{n=1}^... | Thomas | 26,188 | <p>You have
$$
2^{2n}3^{1-n} = (2^2)^n3^{-(n-1)} = 4^n3^{-(n-1)} = \frac{4^n}{3^{n-1}} = \frac{4\cdot4^{n-1}}{3^{n-1}} = 4\left(\frac{4}{3}\right)^{n-1}
$$
Here we have used the formulas</p>
<ol>
<li>$$
a^ma^k = a^{m+k}
$$</li>
<li>$$
a^{-m} = \frac{1}{a^m}
$$</li>
<li>$$
(a^m)^k = a^{mk}
$$</li>
<li>$$
\frac{a^m}{b^m... |
2,836,552 | <blockquote>
<p>Let $R$ be commutative ring with identity that contains a field $K$ as a subring. If$ $R is a finite dimensional vector space over the field $K$, prove that every prime ideal in $R$ will be maximal.</p>
</blockquote>
<p>My idea was to prove the integral domain $R/p$ (if $p$ is a prime ideal) was a fi... | Bernard | 202,857 | <p><strong>Hint</strong>:</p>
<p>Note that $R/\mathfrak p$ has finite dimension over $K$. You have to prove that any non-zero element $x$ in $R/\mathfrak p$ is invertible. Consider the map
$$R/\mathfrak p\xrightarrow {\enspace{}\times x\enspace\:}R/\mathfrak p.$$
This is an endomorphism of the $K$-vector space $R/\m... |
2,836,552 | <blockquote>
<p>Let $R$ be commutative ring with identity that contains a field $K$ as a subring. If$ $R is a finite dimensional vector space over the field $K$, prove that every prime ideal in $R$ will be maximal.</p>
</blockquote>
<p>My idea was to prove the integral domain $R/p$ (if $p$ is a prime ideal) was a fi... | Berci | 41,488 | <p>$A:=R/p$ is also finite dimensional and has no zero divisors. <br>
Take any nonzero $a\in A$ and consider its powers $1,a, a^2,\dots$, these are linearly dependent. Take the least $n\in\Bbb N$ giving $\lambda_na^n+\dots+\lambda_0=0$. </p>
<p>Since we can cancel out $a$, the constant term $\lambda_0$ is nonzero. <br... |
1,206,460 | <p>This is the question :
Prove that the set of all the words in the English language is countble (the set's cardinality is אo)
A word is defined as a finite sequence of letters in the English language.</p>
<p>I'm not really sure how to start this. I know that a finite union of countble sets is countble and i think th... | IanF1 | 187,796 | <p>The easiest way to show a set is countable is to provide a way of counting it - ie a rule to determine the position of any member within the set.</p>
<p>In this case we can start with all 1-letter "words", from a to z - there are 26 of these. Then we can continue with the two-letter words aa, ab, ... az, ba, bb ...... |
2,943,973 | <p>I'm trying to prove there is some <span class="math-container">$N$</span> such that for all <span class="math-container">$n > N$</span>, it is the case that <span class="math-container">$$2n^{3/4} + 2(n-\sqrt{n})^{3/2} + n - 2n^{3/2} \leq 0$$</span></p>
<p>I know that this is true, since I graphed this function ... | Will Jagy | 10,400 | <p><span class="math-container">$$ \sqrt{n - \sqrt n} < \sqrt n - \frac{1}{2} $$</span>
<span class="math-container">$$ \left( n - \sqrt n \right)^{\frac{3}{2}} < n^{\frac{3}{2}} - \frac{3}{2} n + \frac{3}{4} \sqrt n - \frac{1}{8} $$</span></p>
|
1,895,323 | <p>Recently, I had a mock-test of a Mathematics Olympiad. There was a question which not only I but my friends too were not able to solve. The question goes like this: </p>
<p>If,
$$ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{a+b+c} $$<br>
Then what is the value of<br>
$$ \frac{1}{a^5} + \frac{1}{b^5} + \fra... | Asinomás | 33,907 | <p>Options $1$ and $2$ are wrong, take $a=1,b=-1,c=-1$.</p>
<p>Now notice given non-zero real number $a,b,c$ we have:</p>
<p>$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{abc}\overbrace{\iff}^{\text{multiply by a+b+c}} 3+2(ab+bc+ac)=1\iff ab+bc+ac=-1$.</p>
<p>Analogously we have $\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}... |
3,057,819 | <p>I giving a second try to this question. Hopefully, with better problem definition.</p>
<p>I have a circle inscribed inside a square and would like to know the point the radius touches when extended. In the figure A, we have calculated the angle(<code>θ</code>), <code>C</code>(center) , <code>D</code> and <code>E</c... | Patricio | 601,766 | <p>In the case you've drawn, you already know the <span class="math-container">$x$</span> value, assuming the circle has center in <span class="math-container">$(C_x,C_y)$</span> and radius <span class="math-container">$r$</span>, <span class="math-container">$A_x=B_x=C_x+r.$</span> As for the <span class="math-contain... |
2,101,756 | <p>From the power series definition of the polylogarithm and from the integral representation of the Gamma function it is easy to show that:
\begin{equation}
Li_{s}(z) := \sum\limits_{k=1}^\infty k^{-s} z^k = \frac{z}{\Gamma(s)} \int\limits_0^\infty \frac{\theta^{s-1}}{e^\theta-z} d \theta
\end{equation}
The identity ... | user247327 | 247,327 | <p>First, your "Full Question" is <strong>not</strong> the same as the first equation you post. Do you intend one of those "=" to be "-"? Second, is $log_3(y)^5$ untended to be $log(y^5)$? That would fit your "$5log_3(y)$" in your "Full Question". If you mean "$6+ log_3(y)= log_3(y^5)$" that is the same as $6+ log_... |
3,115,347 | <p>Let <span class="math-container">$f:(0,\infty) \to \mathbb R$</span> be a differentiable function and <span class="math-container">$F$</span> on of its primitives. Prove that if <span class="math-container">$f$</span> is bounded and <span class="math-container">$\lim_{x \to \infty}F(x)=0$</span>, then <span class="m... | Peter Foreman | 631,494 | <p>a) The probability that a die does not fall higher than <span class="math-container">$3$</span> is given by
<span class="math-container">$$P(X\le3)=\frac{3}{6}=\frac{1}{2}$$</span>
So as each event is independent we can find the probability that no die falls higher than three by raising this result to the power of <... |
1,064,115 | <p><strong>UPDATE:</strong> Thanks to those who replied saying I have to calculate the probabilities explicitly. Could someone clarify if this is the form I should end up with:</p>
<p>$G_X$($x$) = P(X=0) + P(X=1)($x$) + P(X=2) ($x^2$) + P(X=3)($x^3$)</p>
<p>Then I find the first and second derivative in order to calc... | d125q | 112,944 | <p>This would not be a hypergeometric distribution. You can think of hypergeometric as binomial without replacement, not geometric without replacement (even though the name might suggest otherwise). In other words, hypergeometric doesn't care at which spot the red ball is drawn.</p>
<p>Well, it should be relatively ea... |
1,785,444 | <p>The question says to 'Express the last equation of each system as a sum of multiples of the first two equations." </p>
<p>System in question being: </p>
<p>$ x_1+x_2+x_3=1 $</p>
<p>$ 2x_1-x_2+3x_3=3 $</p>
<p>$ x_1-2x_2+2x_3=2 $</p>
<p>The question gives a hint saying "Label the equations, use the gaussian algor... | Noble Mushtak | 307,483 | <p>NOTE: $r_i$ is the original $i^{th}$ equation as stated in your question above.</p>
<p>Well, let's go through the process of finding the extended echelon form using Gauss-Jordan elimination. Here's the matrix:
$$\left[\begin{matrix}1 \ 1 \ 1 \ 1 \\ 2 \ -1 \ 3 \ 3\\ 1 \ -2 \ 2 \ 2\end{matrix}\right]\left[\begin{matr... |
3,991,691 | <p>I'm having some trouble proving the following:</p>
<blockquote>
<p>Let <span class="math-container">$d$</span> be the smallest positive integer such that <span class="math-container">$a^d \equiv 1 \pmod m$</span>, for <span class="math-container">$a \in \mathbb Z$</span> and <span class="math-container">$m \in \math... | Will Jagy | 10,400 | <p>This sort of thing has a clear description in terms of Conway's topograph; I find it more convenient to use equivalent form <span class="math-container">$u^2 + uv - 5 v^2.$</span> The outcome for the original problem is sequences (note that the rule deals with every other element). For instance,
<span class="math-c... |
3,054,321 | <p>I'm looking for a closed form for this sequence,</p>
<blockquote>
<p><span class="math-container">$$\sum_{n=1}^{\infty}\left(\sum_{k=1}^{n}\frac{1}{(25k^2+25k+4)(n-k+1)^3} \right)$$</span></p>
</blockquote>
<p>I applied convergence test. The series converges.I want to know if the series is expressed with any mat... | Robert Israel | 8,508 | <p>Change the order of summation, so it's <span class="math-container">$\sum_{k=1}^\infty \sum_{n=k}^\infty$</span>.
Then I get
<span class="math-container">$$ {\frac {\zeta \left( 3 \right) \left( 4\,\pi\,\cot \left( \pi/5
\right) -15 \right) }{60}}
$$</span>
You could also write <span class="math-container">$$\cot(... |
1,645,361 | <p>I am aware that the union of subspaces does not necessarily yield a subspace. However, I am confused about the following question: </p>
<blockquote>
<p>(i) Let $U, U'$ be subspaces of a vector space $V$ (both not equal to $V$). Prove that the union of $U$ and $U'$ does not equal $V$.<br>
(ii) Find an example o... | AnalysisStudent0414 | 97,327 | <p>(i) Suppose $V = U \cup U'$. Since the inclusions aren't strict, there is at least one vector in $U'$ not in $U$, and viceversa. Call those $u' \in U'\setminus U$ and $u \in U\setminus U'$. Then, $u+u' \not \in U \cup U'$, but this is absurd since $u+u' \in V$.</p>
<p>(ii) If we take three subspaces, the above reas... |
1,645,361 | <p>I am aware that the union of subspaces does not necessarily yield a subspace. However, I am confused about the following question: </p>
<blockquote>
<p>(i) Let $U, U'$ be subspaces of a vector space $V$ (both not equal to $V$). Prove that the union of $U$ and $U'$ does not equal $V$.<br>
(ii) Find an example o... | Takirion | 299,952 | <p>(i) Let's assume $U,U',V$ as you said, but $U\cup U'=V$. Obviously we have neither $U\subseteq U'$ nor $U'\subseteq U$ as else we had $V=U\cup U'=U$, or $V=U\cup U'=U'$.
So we find $v\in U\setminus U'$ and $v'\in U'\setminus U$. As $U\cup U'=V$ we have $v+v'\in U\cup U'$.
If $v+v'\in U$ we have $v'=(v+v')-v\in U$, c... |
1,657,557 | <p>For example, how would I enter y^(IV) - 16y = 0? </p>
<p>typing out fourth derivative, and putting four ' marks does not seem to work. </p>
| mvw | 86,776 | <p>The expression</p>
<pre><code>solve y'''' - 16 y = 0 for y
</code></pre>
<p>seems to work.</p>
<p><a href="http://www.wolframalpha.com/input/?i=solve+y%27%27%27%27+-+16+y+%3D+0+for+y" rel="nofollow">Link to result</a>.</p>
|
2,129,086 | <p>I know that the total number of choosing without constraint is </p>
<p>$\binom{3+11−1}{11}= \binom{13}{11}= \frac{13·12}{2} =78$</p>
<p>Then with x1 ≥ 1, x2 ≥ 2, and x3 ≥ 3. </p>
<p>the textbook has the following solution </p>
<p>$\binom{3+5−1}{5}=\binom{7}{5}=21$ I can't figure out where is the 5 coming from?</... | Felix Marin | 85,343 | <p>$\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\n... |
897,756 | <p>How can I solve the following trigonometric inequation?</p>
<p>$$\sin\left(x\right)\ne \sin\left(y\right)\>,\>x,y\in \mathbb{R}$$</p>
<p>Why I'm asking this question... I was doing my calculus homework, trying to plot the domain of the function $f\left(x,y\right)=\frac{x-y}{sin\left(x\right)-sin\left(y\right... | Yiorgos S. Smyrlis | 57,021 | <p>$$
\sin(x+h)=\sin x\cos h+\sin h\cos x,
$$
and hence
$$
\frac{\sin(x+h)-\sin x}{h}=\frac{\sin x\cos h+\sin h\cos x-\sin x}{h}=\cos x+\sin x\frac{\cos h -1}{h} \\= \cos x-\sin x\frac{2\sin^2(h/2)}{h}=\cos x-\sin x \cdot \frac{h}{2} \cdot\left(\frac{\sin(h/2)}{h/2}\right)^2 \to \cos x-\sin x\cdot 0 \cdot 1\\=\cos x
$$... |
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